# Selector-Enhancer: Learning Dynamic Selection of Local and Non-local
Attention Operation for Speech Enhancement

Xinmeng Xu1, Weiping Tu1,2,3, Yuhong Yang1,3 This work was supported in part
by the National Nature Science Foundation of China (No. 62071342,
No.62171326), the Special Fund of Hubei Luojia Laboratory (No. 220100019), the
Hubei Province Technological Innovation Major Project (No. 2021BAA034) and the
Fundamental Research Funds for the Central Universities (No. 2042022kf0001).
(Corresponding Author: Weiping Tu).

###### Abstract

Attention mechanisms, such as local and non-local attention, play a
fundamental role in recent deep learning based speech enhancement (SE)
systems. However, natural speech contains many fast-changing and relatively
brief acoustic events, therefore, capturing the most informative speech
features by indiscriminately using local and non-local attention is
challenged. We observe that the noise type and speech feature vary within a
sequence of speech and the local and non-local operations can respectively
extract different features from corrupted speech. To leverage this, we propose
Selector-Enhancer, a dual-attention based convolution neural network (CNN)
with a feature-filter that can dynamically select regions from low-resolution
speech features and feed them to local or non-local attention operations. In
particular, the proposed feature-filter is trained by using reinforcement
learning (RL) with a developed difficulty-regulated reward that is related to
network performance, model complexity, and “the difficulty of the SE task”.
The results show that our method achieves comparable or superior performance
to existing approaches. In particular, Selector-Enhancer is potentially
effective for real-world denoising, where the number and types of noise are
varies on a single noisy mixture.

##  1 Introduction

Speech enhancement (SE) aims at separating the underlying high quality and
intelligently speech s from a noise-corrupted speech signal
$\textbf{y}=\textbf{s}+\textbf{n}$, where n represents additive noise. For
many applications such as mobile communication and hearing aids, SE algorithms
are restricted to operate with low latency and often also to use only single-
channel inputs. With the recent advances in supervised learning, deep neural
networks (DNNs) have become state-of-the-art for single-channel SE. They
typically operate in the short-time Fourier transform (STFT) domain and
estimate the target clean speech from the noisy signal via direct spectral
mapping (Xu et al. 2013) or time-frequency masking (Wang, Narayanan, and Wang
2014).

Several studies have shown the efficiency of local processing by convolutional
neural networks (CNNs) for STFT-domain SE, which are structurally well-suited
to focus on local patterns such as harmonic structures in the speech spectra
(Park and Lee 2017). However, the speech also exhibits non-linguistic long-
term dependencies, such as gender, dialect, speaking rating, and emotional
state (Bengio, Simard, and Frasconi 1994). For aggregating more informative
speech features, several strategies are proposed, such as encoder-decoder
architecture (Park and Lee 2017), dilated convolutions (Pirhosseinloo and
Brumberg 2019) or applying long-short term memory (LSTM) networks (Tan and
Wang 2018; Oostermeijer, Wang, and Du 2021), to enlarge the receptive field.

Figure 1: Comparison with (c) local and non-local attention with feature-
filter, (d) local attention, (e) non-local attention, and (f) local and non-
local attention based SE model enhanced speech spectrogram, in which most
obvious details of spectrogram distinction are framed by a gray and green
dotted box. Note that (a) is a noisy speech spectrogram, and (b) is a clean
speech spectrogram.

Furthermore, the non-local operation (Wang et al. 2018a), which takes the
advantage of self-similarity within the whole speech to recover a local speech
segment, is effective to establish the long-range dependence based on time-
frequency (T-F) units of speech spectra. However, T-F units of noisy speech
are usually corrupted, therefore, establishing the correlation between T-F
units is unreliable, especially when speech is heavily corrupted. Besides, the
non-local operation has $\mathcal{O}(T^{2})$ computational complexity for a
sequence length of $T$, which increases the computational burden for the SE
model, hence the practical usage is limited. In addition to non-local
attention, local attention processes the local neighborhood, which scales more
informative feature regions with “high attention” while putting “low
attention” to other feature regions and achieving a linear complexity to the
input feature. However, local attention is unable to extract speech global
information that is important to speech enhancement tasks, due to the limited
receptive field.

According to the above analysis, local and non-local attentions have
shortcomings and merits. Figure 1(d) and Figure 1(e) show an example enhanced
spectrogram processed by local and non-local attention-based SE models. The
local attention-based speech enhancement model failed to process the noise
component in the green dotted box. In contrast, the non-local attention-based
system insufficiently removes the noise component in the white dotted box.
However, integrating both attentions indiscriminately is a challenge to keep
their merits while avoiding their shortcomings. For example, as shown in
Figure 1(f), the SE system by directly applying local and non-local attention
operations still failed to remove the noise component in the green box while
remaining the low-frequency noise component in the white box.

In this study, we propose Selector-Enhancer, a novel framework that jointly
processes different regions of a noisy speech spectrogram via local and non-
local attention operation with the guidance of adjacent features.
Particularly, a feature-filter is proposed and is trained together with dual-
attention based CNN. The feature-filter first predicts a low-resolution
dispatch policy mask, in which each time-frequency (T-F) unit represents a
policy for a region in the input noisy spectrogram. After that, the features
of different regions are sent to specific attention operations (local or non-
local attention) according to the corresponding policy on the predicted mask.
Since the feature selection is non-differentiable, the feature filter is
trained in a reinforcement learning (RL) framework driven by a reward to
achieve a good complexity-performance trade-off.

In a nutshell, the contributions of our work are three-fold111Code is
available at: https://github.com/XinmengXu/Selector
-Enhancer.

  * •

We propose a novel SE framework for making a trade-off between local and non-
local attention operations by introducing a feature-filter that predicts a
policy mask to select the feature of different spectrogram regions and to send
to specific attention operations.

  * •

We develop a difficulty-regulated reward, which is related to performance,
complexity, and “the difficulty of the SE task”. Specifically, the difficulty
is made by the loss function, since the high loss denotes the high difficulty
of the SE task.

  * •

The proposed Selector-Enhancer achieves comparable and superior performance to
recent approaches with faster speed and is particularly effective when noise
distribution is variant within a speech sequence.

##  2 Related Work

###  2.1 Non-local Attention

Long-range dependencies are important prior to SE, and this prior can be
effectively obtained by non-local attention operation, which calculates the
mutual similarity between T-F units in each frame, which is helpful for
capturing the global information in the frequency domain with a slightly
increased computing complexity. In general, non-local attention is defined as
follows (Wang et al. 2018b):

$y_{i}=\frac{1}{\mathcal{C}(x)}\sum_{\forall j}f(x_{i},x_{j})g(x_{j}),$ (1)

where $x$ and $y$, respectively, represent the input and output tensor of the
operation with the same size, $f$ denotes the pairwise function to calculate
the correlation between the locations of the feature map, $g$ signifies the
unary input function for information transform, and $\mathcal{C}(x)$ is a
normalization factor.

Recently, deep neural networks (DNNs) have been prevalent in the community of
SE, and non-local attention-based SE models have been proposed for modeling
long-term dependencies of speech signals. In (Li et al. 2019a), embedding
function $g$ is a convolutional layer that can be viewed as a linear embedding
function, and dot product, i.e.,
$\phi(\textbf{y}_{i},\textbf{y}_{j})=\textbf{y}_{i}^{\top}\textbf{y}_{j}$ is
applied to compute the similarity between query and keys. However, non-local
attention is unreliable when speech is heavily corrupted and colored noise is
contained in the background mixture. In addition, a transformer with Gaussian-
weighted non-local attention is proposed (Kim, El-Khamy, and Lee 2020), whose
attention weights are attenuated according to the distance between correlated
T-F units. The attenuation is determined by the Gaussian variance which can be
learned during training. While all these methods demonstrate the benefits of
non-local attention, their approach is unreliable when speech is heavily
corrupted and has high computational costs.

Figure 2: The proposed Selector-Enhancer, which is composed of a dual-
attention based CNN and a feature filter. The dual-attention-based CNN
contains $N$ dynamic blocks, each of which has 2 optional paths. The feature
filter is depicted at the bottom left corner. It predicts a policy mask with
two channels, both of which are then up-sampled by nearest-neighbor
interpolation to select eligible regions for the corresponding paths.

###  2.2 Local Attention

Local attention focuses on the important speech components in an audio stream
with “high attention” while perceiving the unimportant region (e.g., noise or
interference) in “low attention”, according to the past and current time
frame, thus adjusting the focal point over time. In (Hao et al. 2019), a
lightweight casual transformer with local self-attention is proposed for real-
time SE in computation resource-limited environments. Local attention based
RNN-LSTM (Oostermeijer, Wang, and Du 2021) is proposed for superior SE
performances by modeling important sequential information, in which the RNN
model learns the weights of past input features implicitly when predicting an
enhanced frame, the attention mechanism calculates correlations between past
frames and the current frame to be enhanced and give weights to past frames
explicitly. Compared with non-local attention, local attention is more
suitable for real-time settings and more lightweight (Hao et al. 2019;
Oostermeijer, Wang, and Du 2021), since they do not rely on future
information. However, compared with non-local operations, the main drawback of
the local attention mechanism during SE is the relatively small receptive
field.

###  2.3 Dynamic Networks

Dynamic networks are effective to achieve a trade-off between speech and
performance in different tasks. Recently, several approaches (Wu et al. 2018;
Shalit, Johansson, and Sontag 2017; Yang et al. 2022) are applied to dynamic
networks to reduce the computational cost in ResNet by dynamically skipping
some residual blocks. Although the aforementioned methods mainly focus on a
single task, a routing network (Rosenbaum, Klinger, and Riemer 2018) is
developed to facilitate multi-task learning, and their method seeks an
appropriate network path for each specific task. Existing dynamic networks
successfully explore the route policy that depends on either the task to
address. However, we aim at selecting different regions of input speech
spectrogram to send to specific attention operations, in which noise
information (noise type and signal-to-noise ratio (SNR)) and speech
information (speaker identity and speech content) have a weighty influence on
the performance and they should be both considered in the selection operation.

##  3 Methodology

Our task is to recover a clean speech s from a noisy observation
$\textbf{y}\in\mathbb{R}^{T\times F\times 2}$. We propose Selector-Enhancer
that can process each noisy observation according to its speech and noise
features through a specific attention mechanism. Figure 2 gives a detailed
illustration of the Selector-Enhancer framework. The Selector-Enhancer takes
the noisy raw waveform and firstly transforms into a short-time Fourier
transform (STFT) spectrogram, denoted by $\textbf{Y}_{\text{in}}$ in Figure 2.
The output of Selector-Enhancer is clean spectrogram
$\textbf{S}_{\text{pred}}$, and then transforms into raw waveform through
inverse STFT.

###  3.1 Architecture of Selector-Enhancer

We aim to design a framework that can offer different attention mechanism
options for SE. To this end, as shown in Figure 2, Selector-Enhancer is
composed of 3 convolutional blocks at the start- and end-point for extracting
features and reconstructing the target speech, and $N$ dynamic blocks in the
middle. Each convolutional block contains a stacked 2D convolutional layer
which is followed by batch normalization, and exponential linear unit (ELU).
Additionally, skip connections are utilized to concatenate the output of each
convolutional block to the input of the corresponding deconvolutional block.

In the $i$-th dynamic block $f^{i}_{DB}$, there is a shared path $f_{S}^{i}$
containing two convolutional blocks for extracting deeper speech features,
which every feature map should pass through. Paralleling the shared path, a
feature-filter generates a probabilistic distribution of plausible paths for
selection by each T-F unit of the input feature map. Following the shared
path, there are two paths for local and non-local attention denoted by
$f^{i}_{LA}$ and $f^{i}_{NA}$. According to the output of the feature-filter,
the path with the highest probability is activated, where the path index is
represented by action $\alpha_{i}$. Since the path selection is for every T-F
unit rather than a feature map, the action $\alpha_{i}$ is a two-channel mask
rather than a scalar. The first channel $m^{i}_{L}$ implies the regions that
go through the path for local attention, and similarly, the second channel
$m^{i}_{N}$ corresponds to the path for non-local attention. The whole process
is formulated as:

$\displaystyle z_{i+1}=f^{i}_{R}$
$\displaystyle(f^{i}_{LA}(f_{S}^{i}(z_{i})\cdot m^{i}_{L})$ (2)
$\displaystyle+f^{i}_{NA}(f_{S}^{i}(z_{i})\cdot m^{i}_{N})+f_{S}^{i}(z_{i})),$

where $z_{i}$ and $z_{i+1}$ represent the input and output of the $i$-th
dynamic block, respectively, and $f^{i}_{R}$ denotes the decoder of dynamic
block. Note that “$\cdot$” denotes the element-wise multiplication, and each
channel of feature $f_{S}^{i}(z_{i})$ performs the same element-wise
multiplication with mask $m^{i}$. Specially, the mask $m^{i}$ utilizes two
branches to generate a 1-D frequency-dimension mask
$\textbf{F}_{M}\in\mathbf{R}^{1\times F}$ and a 1-D time-frame mask
$\textbf{T}_{M}\in\mathbf{R}^{T\times 1}$ in parallel, then combines them with
a matrix multiplication to obtain the final 2-D mask.

Figure 3: Block Diagram of Local Attention.

Figure 4: Block Diagram of Non-local Attention.

###  3.2 Local and Non-local Attention

The local and non-local attention operations are demonstrated as follows:

Local Attention. Inspired by (Li et al. 2019b), we employ a channel-wise
attention mechanism to design the local attention to perform channel selection
with multiple sizes of the receptive field. The detailed architecture of the
local attention is shown in Figure 3. In the local attention, we utilized two
branches to carry different numbers of convolution layers to generate feature
maps with different sizes of receptive filed. The channel-wise attention is
independently performed on these two outputs. As illustrated in Figure 3,
given input feature I, two-layer convolutional blocks and four-layer
convolutional blocks generate their output $\textbf{C}_{2}$ and
$\textbf{C}_{4}$ in a paralleled manner:

$\left\\{\begin{array}[]{lr}\textbf{C}_{2}=f_{2}(\textbf{I})\\\
\textbf{C}_{4}=f_{4}(\textbf{I}),\end{array}\right.$ (3)

where $f_{2}(\cdot)$ and $f_{4}(\cdot)$ refer to functions of the two and four
convolutional blocks. After getting $\textbf{C}_{2}$ and $\textbf{C}_{4}$, the
scaled feature $S$ is obtained in the following steps:

$\left\\{\begin{array}[]{lr}\textbf{v}=\text{GlobalPooling}(\text{Concat}(\textbf{C}_{2}+\textbf{C}_{4}))\\\
\textbf{w}_{2},\textbf{w}_{4}=\text{sigmoid}([f^{1}_{FC}(\textbf{v}),f^{2}_{FC}(\textbf{v})])\\\
\textbf{S}=\textbf{C}_{2}\cdot\textbf{w}_{2}+\textbf{C}_{4}\cdot\textbf{w}_{4},\end{array}\right.$
(4)

where $f^{1}_{FC}(\cdot)$ and $f^{2}_{FC}(\cdot)$ refer to two independent
fully-connected layers. Finally, the final output J can be obtained as
follows:

$\textbf{J}=\text{sigmoid}(f_{conv}(\textbf{S}))\cdot\textbf{I}+\textbf{I},$
(5)

where $f_{conv}$ is the function of one layer convolutional block.

Non-local Attention. The detailed architecture of non-local attention is shown
in Figure 4. Given the input I, we adopt three independent $1\times 1$
convolutional layers with trainable parameters as embedding functions. These
three embedding functions are denoted as $g$, $\theta$, and $\phi$. They are
used to generate the query (Q), key (K), and value (V). The embedding process
does not change the size of the input, which can be defined as follows:

$\left\\{\begin{array}[]{lr}\textbf{Q}=\theta(\textbf{I})\\\
\textbf{K}=\phi(\textbf{I})\\\ \textbf{V}=g(\textbf{I}).\end{array}\right.$
(6)

The distance matrix denoted by A can be efficiently calculated as by dot
product:

$\textbf{A}=\text{softmax}(\textbf{Q}\textbf{K}^{\top}).$ (7)

The output of non-local attention operation W, defined by Eq.(1), is
calculated by the matrix multiplication between A and V. Finally, W is fed
into a $1\times 1$ convolutional operation $o$ to generate the output J.

###  3.3 Feature Filter

The architecture of the feature filter is illustrated in Figure 2, which is
the key component to achieve the selection operation. The feature filter
contains three convolutional blocks, two LSTMs for capturing the correlations
of the feature filter in different dynamic blocks, and a convolutional block
followed by a sigmoid activation function. In addition, the selection
operation is non-differentiable, thus we adopt Markov Decision Process (MDP)
and reinforcement learning (RL) to train the feature filter. The state and
action of this MDP and the developed difficulty-regulated reward for the
feature filter training are clarified as follows:

State and Action. In the $i$-th dynamic block, the state $s_{i}$ include the
input feature $z_{i}$ and the hidden state of LSTM $h_{i}$. Given the state
$s_{i}$, the feature filter produces a distribution of selection that can be
expressed as $f_{AS}=\pi(\textbf{a}|s_{i})$. In the training phase, the action
is sampled from the probabilistic distribution, referred to as
$\textbf{a}_{i}\sim\pi(\textbf{a}|s_{i})$. While in the testing phase, the
action is determined by the highest probability, i.e.,
$\textbf{a}_{i}=\text{argmax}_{a}\pi(\textbf{a}|s_{i})$, in which the action
$\textbf{a}_{i}$ is the two-channel mask that is described in Section 3.1.

Difficulty-regulated Reward. In the RL framework, the feature filter is
trained to maximize a cumulative reward. In this paper, for the attention
selection, we develop a difficulty-regulated reward, which considers the
network performance, complexity, and difficulty of the task. The reward at
$i$-th dynamic block is expressed as:

$r_{i}=\left\\{\begin{array}[]{lr}-\gamma\times\textbf{a}_{i},&1\leq i<N,\\\
-\gamma\times\textbf{a}_{i}+d\times(-\triangle L_{2}),&i=N,\end{array}\right.$
(8)

where $-\gamma$ is the reward penalty for selecting a path in one dynamic
block. In addition, the performance gain in terms of L2 loss. The difficulty
is denoted by $d$ is formulated as:

$d=\left\\{\begin{array}[]{lr}L_{d}/L_{t},&0\leq L_{d}<L_{t},\\\ 1,&L_{d}\geq
L_{t},\end{array}\right.$ (9)

where $L_{d}$ is the Mean Square Error (MSE) loss function and $L_{t}$ is a
threshold. As $L_{d}$ approaches zero, the difficult $d$ decreases, indicating
the input noisy mixture is easy to enhance.

Algorithm 1 REINFORCE algorithm

Get a batch of $K$ noisy-clean speech pairs, $\\{y^{(1)},s^{(1)}\\}$, …,
$\\{y^{(K)},s^{(K)}\\}$

With policy $\pi$, derive ($s_{1}^{(k)}$, $a_{1}^{(k)}$, $s_{1}^{(k)}$, …,
$s_{N}^{(k)}$, $a_{N}^{(k)}$, $s_{N}^{(k)}$)

Compute gradients using REINFORCE algorithm (Williams 1992)

$\triangle\theta=\frac{1}{K}\sum\limits_{k=1}^{K}\sum\limits_{i=1}^{N}\nabla_{\theta}\log\pi(a_{i}^{(k)}|s_{i}^{(k)};\theta)\sum\limits_{j=i}^{N}r_{j}^{(k)}$

Update parameters $\theta\leftarrow\theta+\delta\triangle\theta$

###  3.4 Implementation

The Selector-Enhancer is composed of dual-attention based CNN and feature
filter, and the training process of Selector-Enhancer consists of two stages.
In the first stage, we train a dual-attention based CNN pre-trained model with
a random policy mask. In the second stage, we train the feature filter and
dual-attention based CNN together.

Stage 1. We train the dual-attention based CNN pre-trained model with a
randomly generated policy mask. In this study, we design a loss function by
using MSE loss constrains the output to have reasonable quality.

Stage 2. We train the local and non-local based CNN and feature filter
together. We train the feature filter using the REINFORCE algorithm (Williams
1992), as illustrated in Algorithm 1. The parameters of the feature-filter and
learning rate are denoted by $\theta$ and $\lambda$, respectively.

Figure 5: Visualization of the policy mask and masked feature by feature
filter. Best viewed in color. Table 1: Ablation study of different components
in Selector-Enhancer.

Case Index | 0 (default) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
Local Attention | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Non-local Attention | $\checkmark$ | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Concatenation | $\times$ | $\times$ | $\times$ | $\checkmark$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
Selective Network (Li et al. 2019b) | $\times$ | $\times$ | $\times$ | $\times$ | $\checkmark$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
Feature-Filter (Proposed) | $\checkmark$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Number of DB | 4 | 4 | 4 | 4 | 4 | 4 | 3 | 2 | 1 | 5
PESQ | 3.09 | 2.63 | 2.66 | 2.82 | 2.99 | 2.36 | 3.02 | 2.94 | 2.87 | 3.11
STOI ($\%$) | 93.41 | 87.15 | 87.47 | 91.10 | 92.28 | 78.39 | 92.76 | 91.33 | 91.01 | 93.38
Parameters | 2.37M | 1.98M | 2.27M | 2.38M | 2.55M | 2.15M | 2.24M | 2.03M | 1.81M | 2.57M

##  4 Experiments

###  4.1 Datasets

To evaluate the proposed Selector-Enhancer, we use two types of datasets:
VoiceBank + DEMAND and AVSpeech + AudioSet.

VoiceBank + DEMAND: This is an open dataset proposed by (Yin et al. 2020). The
pre-scaled subset of VoiceBank (Valentini-Botinhao et al. 2016) provided by is
used to train the Selector-Enhancer. We randomly select 40 male speakers and
40 female speakers from the total of 84 speakers (42 male and 42 female), and
each speaker pronounced around 4000 utterances. Therefore, the clean data
contained 320,000 utterances in total (16,000 male utterances and 16,000
female utterances). Next, we mix 32,000 utterances with noise from DEMAND
(Thiemann, Ito, and Vincent 2013) in $\\{-5,-4,-3,-2,-1,0,5\\}$ dB signal-to-
noise ratio (SNR) levels. In addition, we set aside 200 clean utterances from
the training set to create a validation set. Test data are generated by mixing
200 utterances selected from the remained 4 untrained speakers (50 each) with
noise from DEMAND at three SNR levels $\\{-5,0,5\\}$ dB.

AVSpeech+AudioSet: This is a large dataset proposed by (Ephrat et al. 2018).
The clean dataset AVSpeech is collected from Youtube, containing 4700 hours of
video segments with approximately 150,000 distinct speakers spanning a wide
variety of people languages. The noisy speech is a mixture of the above clean
speech segments with AudioSet (Gemmeke et al. 2017) that contains a total of
more than 1.7 million 10-second segments of 526 kinds of noise. The noisy
speech is synthesized by a weighted linear combination of speech segments and
noise segments (Yin et al. 2020):

$\text{Mix}_{i}=\text{Speech}_{j}+0.3\times\text{Noise}_{k},$ (10)

where $\text{Speech}_{j}$ and $\text{Noise}_{k}$ are 4-second segments
randomly sampled from the speech and noise dataset. In our experiment, 10,000
segments were randomly sampled from the AVSpeech dataset for the training set
and 500 segments for the validation dataset. Because of the wide energy
distribution in both datasets, the created noisy speech dataset has a wide
range of SNR.

For the test dataset, we randomly select 840 utterances from 84 speakers in
WSJ0 SI-84 dataset (Paul and Baker 1992) and mingle with unknown types of non-
stationary noises from NOISEX92 (Varga and Steeneken 1993) under three SNR
levels $\\{-5,0,5\\}$ dB. The choice of the training set and test set is to
let the Selector-Enhancer adapt to different speakers and different acoustic
environments.

###  4.2 Experimental Setup

As for the number of dynamic blocks, we use $N=4$ for a fair comparison with
the baseline methods. The threshold of difficulty $L_{t}$ and the reward
penalty $\gamma$ are, respectively, set to $6\times 10^{-2}$ and $8\times
10^{-2}$ based on the specific task. Specifically, when the noise corruption
is severe, the threshold and reward penalty is required to be larger. In the
first training stage, the coefficient of intermediate loss $\beta$ is set to
0.5. All the utterances are sampled at 16 kHz and features are extracted by
using frames of length 512 with frame-shift of 256, and Hann windowing
followed by STFT of size K = 512 with respective zero-padding. We train the
model with a minibatch size of 16. We use Adam as an optimizer and implement
our method on Pytorch.

In our experiments, the performance is assessed by using standard speech
enhancement metrics short-term objective intelligibility (STOI), and
perceptual evaluation of speech quality (PESQ). STOI typically has a value
range from 0 to 1 and can be interpreted as percent correct, and PESQ has a
value range from -0.5 to 4.5.

###  4.3 Ablation Study

We show the ablation study in Table 1 to investigate the effect of different
components in Selector-Enhancer. Note that Case 0 denotes the proposed basic
Selector-Enhancer with the default setting. In the ablation study, we compare
our method with several models: 1) Case 1: we remove all non-local attention
mechanisms, 2) Case 2: we remove all local attention mechanisms, 3) Case 3: We
remove the feature filters and adopt concatenation operation to fuse results
of local and non-local operations, 4) Case 4: we remove feature filters and
adopt selection network (Li et al. 2019b) to fuse results of local and non-
local operations, 5) Case 5: we remove all components of Selector-Enhancer. We
show the detailed analysis of ablation below.

Local Attention Compare with Case 3, Case 2 removes all local attention
mechanisms. According to Table 1, we find that Case 3 improves PESQ by 0.16
and STOI by $3.63\%$ over Case 2. The obvious performance decrease in Case 2
indicates the positive effect of the local attention operation. In addition,
we remove both local and non-local attention mechanisms in Case 5. Case 1
achieves 0.30 PESQ improvement and $9.08\%$ STOI improvement over Case 5. The
performance comparison between Case 1 and Case 5 illustrates the necessity of
the local operation for SE.

Non-local Attention To verify the effect of non-local attention, we also
provide two groups of comparison. Comparing Case 1 and Case 3, one can observe
that local attention mechanisms produce 0.19 PESQ gains and $3.75\%$ STOI
gains together with local attention mechanisms. What is more, From Case 1 to
Case 5, the non-local attention mechanisms contribute 0.27 PESQ gains and
$8.76\%$ STOI gains. Consequently, these two groups of comparison demonstrate
the necessity of the non-local operation for SE.

Attention Collaboration We declare that adopting local and non-local attention
directly by deep learning networks is challenging since the speech contains
dynamic and fast-changing features. To verify the necessity of attention
collaborative operation, we make a comparison between Case 3 and Case 4. Note
that Case 4 incorporates the local and non-local attention mechanisms by
adopting selective network (Li et al. 2019b). From Case 3 to Case 4, the 0.17
PESQ gains and $1.18\%$ STOI gains by Case 4 over Case 3 fully demonstrate the
superiority of the collaborative operation between local and non-local
attention mechanisms.

Feature-Filter To verify the necessity of our proposed feature filter, we
compare Case 0 with Case 3 and Case 4 separately. The 0.27 PESQ gains and
$2.31\%$ STOI gains by Case 0 over Case 3 and the 0.10 PESQ gains and $1.13\%$
STOI gains by Case 0 over Case 4 obviously indicate the effectiveness of our
proposed feature filter for local and non-local attention collaboration.

In order for a better understanding of the feature-filter, we visualize the
masked feature maps processed by the feature filter for local and non-local
attention mechanisms and processed feature maps produced by local and non-
local attention operation in Figure 5. We show that the corrupted speech
components are scaled and masked by feature-filter to separately feed into
local and non-local attention operations.

Number of Dynamic Blocks To study the number of dynamic blocks, we make a
comparison among Cases 0, 6, 7, 8, and 9, which clearly shows that the
performance increases with the number of dynamic blocks. Therefore, for a good
trade-off between speed and accuracy, we use 4 dynamic blocks as our final
proposed model to do comparisons to other state-of-the-art works.

Table 2: Evaluation and comparison of different models on VoiceBank + DEMAND.

SNR | -5 dB | 0 dB | 5 dB | FLOPs (G) | Param. (M) | RTF
---|---|---|---|---|---|---
Metrics | STOI (%) | PESQ | STOI (%) | PESQ | STOI (%) | PESQ
Unprocessed | 56.32 | 1.30 | 70.95 | 1.57 | 81.33 | 1.83 | - | - | -
CRN (2019) | 73.36 | 1.81 | 82.27 | 2.45 | 89.27 | 2.90 | 1.16 | 2.36 | 0.52
CNN-NL (2020) | 77.98 | 1.90 | 85.63 | 2.51 | 90.11 | 3.05 | 5.94 | 5.04 | 0.86
T-GSA (2020) | 80.88 | 2.14 | 88.47 | 2.66 | 91.37 | 3.03 | 6.23 | 5.63 | 0.95
SA-TCN (2021) | 82.47 | 2.26 | 89.89 | 2.84 | 92.73 | 3.25 | 8.85 | 5.81 | 0.91
Selector-Enhancer | 83.38 | 2.36 | 91.24 | 3.03 | 94.36 | 3.37 | 2.19 | 2.37 | 0.63

###  4.4 System Comparison

We carry out the system comparison on both datasets mentioned in section 4.1.

VoiceBank+DEMAND: We train the proposed Selector-Enhancer on the synthetic and
commonly used dataset VoiceBank+DEMAND. The results are provided in Table 2.
We compare our model with 4 recent baselines, CRN (Tan and Wang 2018), CNN-NL
(Li et al. 2019a), T-GSA (Kim, El-Khamy, and Lee 2020), and SA-TCN (Lin et al.
2021). Firstly, the non-local attention based SE models like CNN-NL, T-GSA,
and SA-TCN have a very large gain over the simple CNN-based SE model (CRN) in
terms of STOI and PESQ. This indicates the advantage of non-local attention
operations in capturing long-term dependencies. Also, Selector-Enhancer yields
the best results in all conditions.

Realizing that the number of trainable parameters can not completely reflect
the complexity of a model, especially for deep non-local methods. Thus we
employ the floating-point operations (FLOPs), the number of trainable
parameters, and real-time factor (RTF) to evaluate the complexity of different
methods. The complexity analysis of several representative methods is reported
in Table 2. The proposed Selector-Enhancer has an attractive complexity and
low computational costs and achieves quite promising performance.

Table 3: Performance evaluation scores on AVSpeech + AudioSet

Test SNR | -5 dB | 0 dB | 5 dB
---|---|---|---
Metrics | STOI | PESQ | STOI | PESQ | STOI | PESQ
Mixture | 57.99 | 1.18 | 57.99 | 1.18 | 90.03 | 2.06
U-Net | 72.92 | 1.76 | 87.01 | 2.06 | 91.85 | 2.37
GRN | 74.96 | 1.81 | 88.62 | 2.19 | 91.72 | 2.44
PHASEN | 74.77 | 2.47 | 89.89 | 2.93 | 93.70 | 3.09
TFT-Net | 76.94 | 2.47 | 90.33 | 2.93 | 93.70 | 3.11
Selector-Enhancer | 78.33 | 2.52 | 91.19 | 3.06 | 94.27 | 3.15

AVSpeech+AudioSet: On this large dataset, we compare the Selector-Enhancer
with several state-of-the-art methods,i.e., U-Net (Park and Lee 2017), GRN
(Pirhosseinloo and Brumberg 2019), PHASEN (Yin et al. 2020), and TFT-Net (Tang
et al. 2021). The results in Table 3 demonstrate that the Selector-Enhancer
outperforms these state-of-the-art methods and also indicate that the
Selector-Enhancer can be generalized to various speakers and various kinds of
noisy environments.

(a) STOI score ($\%$)

(b) PESQ score

Figure 6: Comparison in STOI and PESQ average for cases with multiple noise
types, in which B, F, and C denote babble, factory, and cafeteria noisy types,
respectively.

To demonstrate the effectiveness and flexibility of the Selector-Enhancer, we
further explore and apply the Selector-Enhancer to circumstances when multiple
noise types are included in a noisy background. We randomly select 150 clean
utterances from the test set and 3 challenging noises, babble (B), factory
(F), and cafeteria (C) from NOISEX92, which are explained in Section 4.1, to
produce the multiple types of noises with clean utterances at -5 dB, 0 dB, and
5 dB SNR levels. Figure 6 shows the STOI and PESQ results over three mentioned
SNR levels on multiple noise types, where B, F, and C represent babble,
factory, and cafeteria noise types, respectively. According to the Table, we
observe that the results using 1, 2, and 3 noise types in the noisy background
are comparable, although the STOI and PESQ scores are decreasing when the
number of noise types is increase.

##  5 Conclusion

In this paper, we propose Selector-Enhancer that enables attention selection
for each speech feature region. Specifically, Selector-Enhancer contains a
dual-attention based CNN and a feature filter. The dual-attention based CNN
offers options for local and non-local attention mechanisms to address
different types of noisy mixtures while the feature-filter selects the optimal
regions of speech spectra to feed into the specific attention mechanisms.
Comprehensive ablation studies have been carried out, justifying almost every
design choice we have made in Selector-Enhancer. Experimental results show
that the proposed method is capable of superior performance in SE by
outperforming the state-of-the-art methods with fewer computational costs.

## References

  * Bengio, Simard, and Frasconi (1994) Bengio, Y.; Simard, P.; and Frasconi, P. 1994.  Learning long-term dependencies with gradient descent is difficult.  _IEEE transactions on neural networks_ , 5(2): 157–166.
  * Ephrat et al. (2018) Ephrat, A.; Mosseri, I.; Lang, O.; Dekel, T.; Wilson, K.; Hassidim, A.; Freeman, W. T.; and Rubinstein, M. 2018.  Looking to listen at the cocktail party: a speaker-independent audio-visual model for speech separation.  _ACM Transactions on Graphics (TOG)_ , 37(4): 1–11.
  * Gemmeke et al. (2017) Gemmeke, J. F.; Ellis, D. P.; Freedman, D.; Jansen, A.; Lawrence, W.; Moore, R. C.; Plakal, M.; and Ritter, M. 2017.  Audio set: An ontology and human-labeled dataset for audio events.  In _2017 IEEE international conference on acoustics, speech and signal processing (ICASSP)_ , 776–780. IEEE.
  * Hao et al. (2019) Hao, X.; Shan, C.; Xu, Y.; Sun, S.; and Xie, L. 2019.  An attention-based neural network approach for single channel speech enhancement.  In _ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 6895–6899. IEEE.
  * Kim, El-Khamy, and Lee (2020) Kim, J.; El-Khamy, M.; and Lee, J. 2020.  T-gsa: Transformer with gaussian-weighted self-attention for speech enhancement.  In _ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 6649–6653. IEEE.
  * Li et al. (2019a) Li, X.; Li, Y.; Li, M.; Xu, S.; Dong, Y.; Sun, X.; and Xiong, S. 2019a.  A Convolutional Neural Network with Non-Local Module for Speech Enhancement.  In _Interspeech_ , 1796–1800.
  * Li et al. (2019b) Li, X.; Wang, W.; Hu, X.; and Yang, J. 2019b.  Selective kernel networks.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 510–519.
  * Lin et al. (2021) Lin, J.; van Wijngaarden, A. J. d. L.; Wang, K.-C.; and Smith, M. C. 2021.  Speech enhancement using multi-stage self-attentive temporal convolutional networks.  _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , 29: 3440–3450.
  * Oostermeijer, Wang, and Du (2021) Oostermeijer, K.; Wang, Q.; and Du, J. 2021.  Lightweight Causal Transformer with Local Self-Attention for Real-Time Speech Enhancement.  In _Proc. Interspeech 2021_ , 2831–2835.
  * Park and Lee (2017) Park, S. R.; and Lee, J. W. 2017.  A Fully Convolutional Neural Network for Speech Enhancement.  _Proc. Interspeech 2017_ , 1993–1997.
  * Paul and Baker (1992) Paul, D. B.; and Baker, J. 1992.  The design for the Wall Street Journal-based CSR corpus.  In _Speech and Natural Language: Proceedings of a Workshop Held at Harriman, New York, February 23-26, 1992_.
  * Pirhosseinloo and Brumberg (2019) Pirhosseinloo, S.; and Brumberg, J. S. 2019.  Monaural Speech Enhancement with Dilated Convolutions.  In _INTERSPEECH_ , 3143–3147.
  * Rosenbaum, Klinger, and Riemer (2018) Rosenbaum, C.; Klinger, T.; and Riemer, M. 2018.  Routing Networks: Adaptive Selection of Non-Linear Functions for Multi-Task Learning.  In _International Conference on Learning Representations_.
  * Shalit, Johansson, and Sontag (2017) Shalit, U.; Johansson, F. D.; and Sontag, D. 2017.  Estimating individual treatment effect: generalization bounds and algorithms.  In _International Conference on Machine Learning_ , 3076–3085. PMLR.
  * Tan and Wang (2018) Tan, K.; and Wang, D. 2018.  A Convolutional Recurrent Neural Network for Real-Time Speech Enhancement.  In _Interspeech_ , volume 2018, 3229–3233.
  * Tang et al. (2021) Tang, C.; Luo, C.; Zhao, Z.; Xie, W.; and Zeng, W. 2021.  Joint time-frequency and time domain learning for speech enhancement.  In _Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence_ , 3816–3822.
  * Thiemann, Ito, and Vincent (2013) Thiemann, J.; Ito, N.; and Vincent, E. 2013.  The Diverse Environments Multi-channel Acoustic Noise Database (DEMAND): A database of multichannel environmental noise recordings.  In _Proceedings of Meetings on Acoustics ICA2013_ , volume 19, 035081\. Acoustical Society of America.
  * Valentini-Botinhao et al. (2016) Valentini-Botinhao, C.; Wang, X.; Takaki, S.; and Yamagishi, J. 2016.  Investigating RNN-based speech enhancement methods for noise-robust Text-to-Speech.  In _SSW_ , 146–152.
  * Varga and Steeneken (1993) Varga, A.; and Steeneken, H. J. 1993.  Assessment for automatic speech recognition: II. NOISEX-92: A database and an experiment to study the effect of additive noise on speech recognition systems.  _Speech communication_ , 12(3): 247–251.
  * Wang et al. (2018a) Wang, X.; Girshick, R.; Gupta, A.; and He, K. 2018a.  Non-local neural networks.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 7794–7803.
  * Wang et al. (2018b) Wang, X.; Girshick, R.; Gupta, A.; and He, K. 2018b.  Non-local neural networks.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , 7794–7803.
  * Wang, Narayanan, and Wang (2014) Wang, Y.; Narayanan, A.; and Wang, D. 2014.  On training targets for supervised speech separation.  _IEEE/ACM transactions on audio, speech, and language processing_ , 22(12): 1849–1858.
  * Williams (1992) Williams, R. J. 1992.  Simple statistical gradient-following algorithms for connectionist reinforcement learning.  _Machine learning_ , 8(3): 229–256.
  * Wu et al. (2018) Wu, Z.; Nagarajan, T.; Kumar, A.; Rennie, S.; Davis, L. S.; Grauman, K.; and Feris, R. 2018.  Blockdrop: Dynamic inference paths in residual networks.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , 8817–8826.
  * Xu et al. (2013) Xu, Y.; Du, J.; Dai, L.-R.; and Lee, C.-H. 2013.  An experimental study on speech enhancement based on deep neural networks.  _IEEE Signal processing letters_ , 21(1): 65–68.
  * Yang et al. (2022) Yang, C.-H. H.; Hung, I.-T. D.; Ouyang, Y.; and Chen, P.-Y. 2022.  Training a resilient q-network against observational interference.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 36, 8814–8822.
  * Yin et al. (2020) Yin, D.; Luo, C.; Xiong, Z.; and Zeng, W. 2020.  Phasen: A phase-and-harmonics-aware speech enhancement network.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, 9458–9465.


* [48] J.J. Blanco-Pillado, K. Sousa and M.A. Urkiola, _Slepian models for Gaussian Random Landscapes_ , _JHEP_ 05 (2020) 142 [1911.07618].
  * [49] S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri and M. Schillo, _Emergent de Sitter Cosmology from Decaying Anti–de Sitter Space_ , _Phys. Rev. Lett._ 121 (2018) 261301 [1807.01570].
  * [50] J.R. Espinosa, _A Fresh Look at the Calculation of Tunneling Actions_ , _JCAP_ 07 (2018) 036 [1805.03680].
  * [51] J.R. Espinosa, _Fresh look at the calculation of tunneling actions including gravitational effects_ , _Phys. Rev. D_ 100 (2019) 104007 [1808.00420].
  * [52] S. Lanza, F. Marchesano, L. Martucci and I. Valenzuela, _Swampland Conjectures for Strings and Membranes_ , _JHEP_ 02 (2021) 006 [2006.15154].
  * [53] S. Lanza, F. Marchesano, L. Martucci and I. Valenzuela, _The EFT stringy viewpoint on large distances_ , _JHEP_ 09 (2021) 197 [2104.05726].
  * [54] G.F. Casas, F. Marchesano and D. Prieto, _Membranes in AdS 4 orientifold vacua and their Weak Gravity Conjecture_, _JHEP_ 09 (2022) 034 [2204.11892].
  * [55] H. Ooguri and C. Vafa, _Non-supersymmetric AdS and the Swampland_ , _Adv. Theor. Math. Phys._ 21 (2017) 1787 [1610.01533].
  * [56] E. Witten, _Instability of the Kaluza-Klein Vacuum_ , _Nucl. Phys. B_ 195 (1982) 481.
  * [57] J.J. Blanco-Pillado and B. Shlaer, _Bubbles of Nothing in Flux Compactifications_ , _Phys. Rev. D_ 82 (2010) 086015 [1002.4408].
  * [58] J.J. Blanco-Pillado, H.S. Ramadhan and B. Shlaer, _Decay of flux vacua to nothing_ , _JCAP_ 10 (2010) 029 [1009.0753].
  * [59] J.J. Blanco-Pillado, H.S. Ramadhan and B. Shlaer, _Bubbles from Nothing_ , _JCAP_ 01 (2012) 045 [1104.5229].
  * [60] M. Dine, P.J. Fox and E. Gorbatov, _Catastrophic decays of compactified space-times_ , _JHEP_ 09 (2004) 037 [hep-th/0405190].
  * [61] G.T. Horowitz, J. Orgera and J. Polchinski, _Nonperturbative Instability of AdS(5) x S**5/Z(k)_ , _Phys. Rev. D_ 77 (2008) 024004 [0709.4262].
  * [62] A.R. Brown and A. Dahlen, _Bubbles of Nothing and the Fastest Decay in the Landscape_ , _Phys. Rev. D_ 84 (2011) 043518 [1010.5240].
  * [63] H. Ooguri and L. Spodyneiko, _New Kaluza-Klein instantons and the decay of AdS vacua_ , _Phys. Rev. D_ 96 (2017) 026016 [1703.03105].
  * [64] I.n. García Etxebarria, M. Montero, K. Sousa and I. Valenzuela, _Nothing is certain in string compactifications_ , _JHEP_ 12 (2020) 032 [2005.06494].
  * [65] P. Draper, I.G. Garcia and B. Lillard, _Bubble of nothing decays of unstable theories_ , _Phys. Rev. D_ 104 (2021) L121701 [2105.08068].
  * [66] P. Draper, I. Garcia Garcia and B. Lillard, _De Sitter decays to infinity_ , _JHEP_ 12 (2021) 154 [2105.10507].
  * [67] J.J. Blanco-Pillado, B. Shlaer, K. Sousa and J. Urrestilla, _Bubbles of Nothing and Supersymmetric Compactifications_ , _JCAP_ 10 (2016) 002 [1606.03095].
  * [68] I.L. Buchbinder and S.M. Kuzenko, _Ideas and methods of supersymmetry and supergravity or a walk through superspace_ , Institute of Physics Publishing (1998).
  * [69] H. Luckock and I. Moss, _The Quantum Geometry of Random Surfaces and Spinning Membranes_ , _Class. Quant. Grav._ 6 (1989) 1993.
  * [70] I.G. Moss, _Boundary terms for eleven-dimensional supergravity and M theory_ , _Phys. Lett. B_ 577 (2003) 71 [hep-th/0308159].


# On Prime Numbers and The Riemann Zeros

Lucian M. Ionescu  Department of Mathematics, Illinois State University, IL
61790-4520<EMAIL_ADDRESS>
###### Abstract.

The current research regarding the Riemann zeros suggests the existence of a
non-trivial algebraic/analytic structure on the set of Riemann zeros [1, 2,
3]. The duality between primes and Riemann zeta function zeros suggests some
new goals and aspects to be studied: adelic duality and the POSet of prime
numbers.

The article presents computational evidence of the structure of the imaginary
parts $t$ of the non-trivial zeros of the Riemann zeta function $\rho=1/2+it$,
called in this article the Riemann Spectrum, using the study of their
distribution, as in [3].

The novelty represents in considering the associated characters $p^{it}$,
towards an algebraic point of view, than rather in the sense of Analytic
Number Theory. This structure is tentatively interpreted in terms of adelic
characters, and the duality of the rationals.

Second, the POSet structure of prime numbers studied in [4], is tentatively
mirrored via duality in the Riemann spectrum. A direct study of the
convergence of their Fourier series, along Pratt trees, is proposed.

Further considerations, relating the Riemann Spectrum, adelic characters and
distributions, in terms of Hecke idelic characters, local zeta integrals
(Mellin transform) and $\omega$-eigen-distributions, are explored following
[11].

##  1\. Introduction

One of the most famous and important problem in mathematics is the Riemann
Hypothesis (RH), not only because of the multitude of conditional theorems
depending on its validity, but also because of the connections with
multiplicative number theory and related areas of mathematics.

Deeply connected to mathematical-physics, e.g. with Riemann gas, quantum
chaos, random matrices etc. (see also [5]), the zeta function zeroes are
“dual” with the prime numbers [21, 1], at least in the sense that the Riemann-
Mangoldt exact formula resembles a Poisson summation formula. Understanding
this duality would clarify the “origin” of these zeroes, beyond our capability
to compute them and count them precisely, perhaps shedding light on the “right
approach” to prove the RH.

In a different direction, a natural partial order on the set of prime numbers
(POSET) was recently (re)discovered [4], being related to the concept of Pratt
tree [6] (1975) used in primality tests. It is therefore natural to ponder on
a possible mirroring phenomenon, as a yet to be determined structure on the
set of Riemann zeros. The present article is a preliminary investigation in
this direction, exploring the adelic duality as a theoretical avenue
supporting the existence of such a structure, and its relation to the POSet of
primes.

In what follows, the imaginary parts of the non-trivial zeros of the Riemann
zeta function are called the Riemann spectrum [21] or zeta eigenvalues
(anticipating a relation with the adelic $\omega$-eigendistributions).

On the side of the prime numbers, the author introduced a partial order
derived conceptually from the internal symmetry structure of basic finite
fields [4]: $q<<p$ if $q|p-1$, i.e. $Aut_{Ab}(Z/pZ,+)$ has a $q$-order
symmetry (generator). This leads to a hierarchy on primes, with a rooted tree
associated to each prime.

For example

$t(181)=B^{+}(\bullet,t(2),t(2),t(3),t(3),t(5)),\ t(3)=B^{+}(\bullet),\
t(5)=B^{+}(\bullet,\bullet).$

$\textstyle{t(181)=}$$\textstyle{\circ\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi(181)=2^{2}\cdot
3^{2}\cdot
5}$$\textstyle{\overset{2}{\bullet}}$$\textstyle{\overset{2}{\bullet}}$$\textstyle{\overset{3}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{3}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overset{5}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi(3)=2,\
\phi(5)=2^{2}}$$\textstyle{\overset{2}{\bullet}}$$\textstyle{\overset{2}{\bullet}}$$\textstyle{\overset{2}{\bullet}}$$\textstyle{\overset{2}{\bullet}}$

where $B^{+}$ is the operator which adjoins a common root to the input rooted
trees. Of course the labels ($2,3$ and $5$) are not necessary; the
corresponding “prime values” of the labels can be reconstructed recursively,
from terminals going up.

In this way the POSet of primes is a connected directed graph, with no
directed cycles, defining a “flow” with sink at $2$.

On the other side of the primes-zeros duality, some progress was achieved in
the understanding of the imaginary part of Riemann zeta function zeros (ZZ)
$\rho_{n}=1/2+2\pi i\gamma_{n}$, assuming of course the Riemann Hypothesis is
true. Namely, in [3, 7] the set of imaginary parts of (non-trivial) zeta
zeroes $\gamma=\\{\gamma_{n}\\}_{n}$ is investigated by looking at the changes
under multiplication by real numbers
$L_{\alpha}(\gamma_{n})=\\{\alpha\gamma_{n}\\}$, after reduction to the unit
interval ($\\{\ \\}$ denotes the fractional part).

The work of [3], combined with the idea of duality and the additional POSet
structure on primes, suggests the existence of additional structures on them.
More precisely, since the duality is of adelic origin, with a Fourier
transform correspondence between primes and zeros, a similar POSet structure
is expected to exist on them.

We will present next, in more detail, the current known results regarding the
distribution of the Riemann spectrum.

Our reinterpretation of their direct study of the imaginary parts, in terms of
characters $p^{it}$, opens an avenue for investigation using Gauss and Jacobi
sums, with a possibility to relate them to the Weil zeros, in a follow-up
paper. In this present study we will focus on exploring the adelic
interpretation and the relation with the POSet structure of the prime numbers.

##  2\. The Distribution of the Riemann Spectrum

It becomes more and more evident that not the zeros $\rho_{n}$ are the
fundamental, but their imaginary parts $\gamma_{n}$, with $n^{-1/2}$ a
normalization factor, analog to the one present in the discrete Fourier
transform for finite Abelian groups $Z/nZ$.

It is known that the fractional parts of $\\{\gamma_{n}\\}$ are uniformly
distributed on the unit interval [7], p.1. This result can be interpreted as
dual to Dirichlet Theorem regarding the uniform distribution of prime numbers
on the “finite circles” $Z/pZ$. Here the result refers to the Riemann zeros
instead, and the finite cycle $Z/pZ$ is replaced by the normalization of the
unit circle $R/Z\cong T=S^{1}$.

This approximation result is formulated as the integration pairing with a
function of an appropriate class [7], p.2, Eq. (1.1):

(1) $\sum_{0<\gamma\leq T}f(\alpha\gamma)=N(T)\int_{T}f(x)dx+o(N(T),$ (2)
$N(T)=\frac{T}{2\pi}(\log\frac{T}{2\pi}-1)+O(logT).$

Ford and Zaharescu found a lower order term of the approximation present only
when $\alpha=a/q\cdot\log p/2\pi$ with $r=a/q$ rational

$\sum_{0<\gamma\leq
T}f(\alpha\gamma)=N(T)\int_{T}f(x)dx+T\int_{T}f(x)g_{\alpha}(x)dx+o(T)$

where:

(3) $g_{\alpha}(x)=-\frac{\log p}{\pi}Re[\sum_{1}^{\infty}\frac{e^{-2\pi
iqkx}}{p^{ak/2}}].$

It is important to note that re-shuffling the imaginary parts by an irrational
scaling factor $\alpha\ni R$ keeps their distribution on the unit interval
uniform (and $g_{\alpha}\cong 0$).

We interpret their result as a clear indication of a hidden structure on the
imaginary parts of zeta zeros.

######  Remark 2.1.

We also compare their approximation formula for counting zeros with the well-
known Riemann-Mangoldt exact formula for the counting of primes, with emphasis
on the sum over Riemann zeros

$\sum_{\rho}x^{\rho}/\rho+\sum_{1}^{\infty}x^{-2k}/(-2k)=x-\sum_{n\leq
x}\Lambda(n)-\zeta^{\prime}(0)/\zeta(0).$

as in the old formula of Landau ([3], p.1):

$\sum_{0<\gamma\leq T}x^{\rho}=-\frac{T}{2\pi}\Lambda(x)+O(T).$

The presupposed symmetry between primes and zeros, warranted by this yet
unknown explicit duality, should reflect in an analog of the RM-exact formula,
yielding a count of zeros, with an appropriate measure, in terms of the prime
numbers.

Note that characteristic functions of intervals in $T$ are among the
admissible functions $f$ [7], p.2., in the above distributional approximation
formula (weak convergence of measures).

Our goal is to find a hypothetical “direct sum” decomposition on the set of
zeta zeros, as eigenvalues belonging to different “sectors” of the regular
adelic representation. This would be a sort of “localization”, a reflection of
a local-to-global principle applied to the zeros themselves.

To get a feel of the above “resonance” effect (role of an eigenvalue), of
shuffling the zeros via scaling, we reproduce the distributions from [3],
Fig.1, p.4.

For $p=2$ and therefore $\alpha=\log 2/2\pi$, the distribution of
$\gamma_{n}$, i.e. of the imaginary parts of the Riemann zeros modulo $2\pi$
and rescaled to the unit interval, has a pick at $0.5$.

Figure 1. $p=2,\ q=1$

Similar non-trivial distributions occur for $p=5$ and $q=3$ (interpreted as a
compression factor), shown in Fig.2.

Figure 2. $p=5,\ q=3$

On the other hand when mixed using a composite number $p=6$, the uniform
distribution is obtained in Fig.3.

Figure 3. $p=6,\ q=1$

Before explaining further in §5 the significance of these diagrams, we will
briefly review the some facts regarding adelic duality and Fourier transform
at the level of distributions.

##  3\. On Quasi-crystals and Fourier Transform

Fourier transform relating the $\delta$-basis of primes in $Q$ with a dual
basis of characters of $\hat{Q}$, is reminiscent of the old hunch of crystal
basis [19], also suggested by the Odlyzko’s computation [20], as reported by
Baez [18].

###  3.1. Quasicrystals

The fact observed by [20] (1989) and rediscovered my Matt McIrvin around
1990s, is that summing exponentials $exp(i\gamma_{n}x)$ yields an
approximation of the distribution ${\mathcal{D}}_{P}=\sum_{p\in P,n\in N}\log
p\ \delta_{\log p^{n}}$.

Figure 4. From Baez [18], p.4

In the following probabilistic framework:

$\textstyle{(g_{P},d\mu)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Exp}$$\scriptstyle{iso}$$\scriptstyle{\int}$$\textstyle{(N^{\times},\cdot,dn)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ln(n)}$$\textstyle{d\mu=\log(p)dp}$$\textstyle{(R,+,dx)}$$\textstyle{Exp(\sum
k_{p}X_{p})=\prod_{p}p^{k_{p}}.}$

with the “exponential” an isomorphism of groups which allows to interpret the
Chebyshev’s function as an integral ($\psi(n)=\int_{P}k_{n}(p)d\mu(p)$), the
basis of primes in $g_{P}=ZP$ (the corresponding $Z$-module) has a natural
weight (measure) $\log p$, so it is natural to consider the measure
${\mathcal{M}}_{P}=\sum_{p}\log p\ \delta_{p}$ as the counting measure
corresponding to ${\mathcal{D}}_{P}$.

###  3.2. Fourier Transform of Distributions

The duality between primes and zeta zeros is presently known at the level of
Fourier transform of distributions, of the corresponding characteristic sets
[21], Part IV.

With our adapted notation (loc. cit.: $T\leftrightarrow H,P=\Psi$), we have:

$\textstyle{T(t)=\sum\limits_{n=0}^{\infty}\cos(\theta_{n}t)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
w-lim}$$\textstyle{{\mathcal{D}}_{P}(t)=\iint\limits_{P\times
N}\frac{1}{\sqrt{p}^{n}}d_{\log
p^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{FT}$$\scriptstyle{Duality}$$\textstyle{P(\theta)=\mathop{\sum\sum}\limits_{(p,n)\in
P\times N}\frac{\log p}{\sqrt{p}^{n}}\cos(n\log p\
\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad
w-lim}$$\textstyle{{\mathcal{D}}_{\Theta}(\theta)=\sum\limits_{n=0}^{\infty}d_{\theta_{n}}(\theta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $d_{x}=1/2(\delta_{-x}+\delta_{x})$ is the symmetrized Dirac delta
distribution, so that $\hat{d_{n}}=\cos(x)$, and the $(P,d\mu)$ measure space
includes the weight $\log p$ in the measure.

The diagonal arrows represent taking the Fourier transform of distributions,
while the weak limits $w-\lim$ correspond to comparing the “peaks” of finite
approximations with the appropriate delta functions, as explained below.

####  3.2.1. From Primes to Zeros

The following diagram represents an approximation of the Fourier transform
$\hat{{\mathcal{D}}_{P}^{C}}=\sum_{p,n,p^{n}<C}\frac{\log
p}{\sqrt{p}^{n}}d_{\log p^{n}}(t)$, using a cut-off $C=3$ (similarly
$P_{C}(\theta)$ is the partial sum with terms $p^{n}\leq C$).

Figure 5. From [21], part IV, p.107

The peaks of the Fourier Transform seem to converge to the zeta zeros
$\theta_{n}$ (see Fig.2). In the distributional sense:

$\lim_{C\to\infty}\hat{{\mathcal{D}}_{P}^{C}}(\theta)=\lim_{C\to\infty}\sum_{p^{n}<C}\frac{\log
p}{\sqrt{p}^{n}}\cos(n\log p\
\theta)=\sum_{n}d_{\theta_{n}}(\theta)={\mathcal{D}}_{\Theta}(\theta).$

####  3.2.2. From Zeros to Primes

In the other direction, from the Riemann spectrum to the primes, the Inverse
Fourier Transform yields in a similar way, via approximations (see Fig.3):

$\hat{{\mathcal{D}}_{\Theta}}(\log s)=\sum_{n}\cos(\theta_{n}\log
s)\overset{w}{=}\sum_{p\in P,n\in N}\log p\ d_{p^{n}}(s)=\iint\limits_{P\times
N}d_{p^{n}}(s).$

Figure 6. From [21], part IV, p.117

The maxima of the $C=1000$ approximation $T_{C}(\theta$ are localized at power
of primes, the limit being ${\mathcal{D}}_{P}\circ e^{t}$. With $s=e^{t}$ we
obtain $\hat{{\mathcal{D}}_{\Theta}}={\mathcal{D}}_{P}$, as expected.

##  4\. The Adelic and Idelic Characters

Since Tate’s thesis [11], the modern tool to understand analytic continuation
and functional equations of L-functions proved to be harmonic analysis on the
adeles; to understand their instrumental role, a brief review of adelic
(additive) and idelic (multiplicative) characters is in order.

It is well known that the finite primes analog of Pontryagin duality $Z\to
R\to T$ is $Q\to A\to\hat{Q}$, where $Q=(Q,+)$ is viewed as an additive group,
$A=A_{Q}$ denotes the adeles of the field $Q$, and $\hat{Q}$ is the dual group
of additive characters [9] ([10], Ch.3).

The additive characters decompose as a product over the places of $Q$ (its
completions):

$\psi(x)=\prod\psi_{p}(x),\ \psi_{\infty}(x,y)=e^{2\pi i<x,y>_{\infty}},x,y\in
R\ \psi_{p}(a,b)=e^{<a,b>_{p}},a,b\in A,$ $<,>:A\times A^{*}\to T,\
<a,b>=\sum_{q\in\bar{P}}\\{a_{q}b^{*}_{q}\\}_{p},\ a,b\in A$

where $<,>$ is the “Hermitean inner product” on adeles, “gluing” $A$ and its
Hermitean dual $A^{*}$, in analogy with the Hilbert space
formalism111Reinterpreting a Hermitean inner product as a bilinear mapping, by
conjugating scalar multiplication on the dual Hilber space.,
$\bar{P}=P\cup\\{\infty\\}$ is the set of places of $Q$, finite in infinite,
while $\\{a_{p}\\}_{p}:Q_{p}\to Q_{p}/Z_{p}$ is the principal part of the
p-adic component $a_{p}$ of the adele $a$. Since there are only finitely many
components with non-trivial principal parts, the sum if finite. It is viewed
as an element of the adelic infinite torus

$T=R/Z\oplus\prod_{p\in P}Q_{p}/Z_{p}\cong S^{1}\oplus Q/Z.$

Note that the orientation on the rational circle $Q/Z$ corresponds with the
opposite orientation on the real circle $R/Z$, under the Hermitean conjugation
on the adeles. It ensures the kernel of the character map $a\to\psi_{a}(b)$ is
precisely the diagonal rationals. It also corresponds to Artin’s product
formula of norms, archimedian and non-archimedian (product being 1).

######  Remark 4.1.

The additive characters of the rational adeles are given by the following
composition

$\psi(a,b)=exp\circ\\{\ \\}\circ m_{A}:A\times A\to A\to T\to S^{1},$

where $\\{\ \\}$ is the projection onto the principal part and $m_{A}$ is the
componentwise multiplication. They form an affine space with structure space
the multiplicative “translations” $L_{a}(b)=ab$:

$\psi_{a}\circ L_{b}=\psi_{ab}.$

Their restrictions to $Q$ yield the additive characters of $Q$.

The multiplicative characters of $Q^{\times}$ are the obvious ones, from its
structure as a free abelian group with basis $P$. The adelic multiplicative
characters $\omega:A^{\times}/Q^{\times}$, trivial on $Q^{\times}$, are called
Hecke characters. They have the form $\omega(x)=\chi(x)|x|^{s}$, with $\chi$ a
Dirichlet character. The finite order Hecke characters are the Dirichlet
characters.

With the adeles as the natural framework for a prime-zeros duality, a natural
question arises: “Is there a p-adic decomposition of the Riemann Spectrum, a
“homological” analog of the Euler products of the L-functions?”

##  5\. In Search of p-adic Sectors of the Riemann Spectrum

Returning to the findings from [3], consider the reals acting on the imaginary
part of the zeros $\gamma$ via dilations $L_{\alpha}(x)=\alpha\gamma$, and
then plotting the distribution of their fractional (principal) part.

When $\alpha$ is real the distribution is unchanged: uniform.

If $\alpha=a\log p/2\pi q$ ($a/q$ rational) the distribution $g_{\alpha}(x)$
(Equation 3) has global minima at the points $k/q,k=0,...,q-1$, corresponding
to the $q$-roots of unity ([3], p.3):

$g_{\alpha}(k/q)=\frac{\log p}{\pi(1-\sqrt{p}^{a})}<0,\ k=0,...,q-1.$

This means “there is a shortage of zeros ” with imaginary parts having the
fractional part close to $k/q$.

###  5.1. Relation to Adeles: Start-up Conjectures

This suggests a natural working hypothesis. But first, some handy terminology:

######  Definition 5.1.

The imaginary parts of the Riemann zeta zeros are called the zeta eigenvalues,
and denoted $\lambda_{n}=\gamma_{n}/(2\pi)$.

######  Conjecture 5.1.

The zeta eigenvalues are “adelic in nature” corresponding to prime numbers
(adelic sectors), while the trivial zeros correspond to the infinite prime
Euler factor 222When choosing the Gaussian distribution as a test function, it
involves the gamma function [13].

The second part is compatible with the known structure of the zeta function:

$\zeta=\xi\cdot 1/\Gamma$

with $\xi$ the complete zeta function, corresponding to the non-trivial zeros
[13].

######  Remark 5.1.

This decomposition should be interpreted in the context of zeta integrals of
the rational adeles, which have a direct product decomposition in terms of
local zeta integrals [11].

In order to better understand this formula, we recast it in a slightly
different form.

First recall that $A\cong Q+[0,1)\times\prod Z_{p}$. Also the correspondence
between $[0,1)$ ($R/Z$) and $S^{1}$ ($p$-roots of unity $Q_{p}/Z_{p}$; recall
$Q/Z\cong\prod Q_{p}/Z_{p}$) requires a normalization by $2\pi$.

So, introduce the variable $\lambda=\gamma/(2\pi)$, suggested by the zeros
count formula Equation 2.

######  Remark 5.2.

We take a moment to comment on its interpretation of a quantity of
information.

$N(2\pi x)=x\log x-x+O(\log x)=\int_{0}^{x}\log tdt+O(\log x)$

is reminiscent of a quantity of information of the distribution of zeta
eigenvalues In the case of a finite probability distribution $Z=\\{p_{n}\\}$
($\sum p_{n}=1$ are probabilities, not prime numbers):

$I(x)=\sum_{p_{n}<x}\log p_{n},H(x)=\sum_{p_{n}<x}p_{n}\log
p_{n}\overset{x\to\infty}{\to}H(Z)\ Shannon\ entropy.$

This is compatible with the interpretation of primes as energy levels in the
Riemann gas model, except it refers to zeta eigenvalues!

This formula counting the number of zeta zeros should have an “exact form”
analog to the Riemann-Mangoldt explicit / exact formula for its “dual”
counting function $\pi(x)$. This suggests to look at a similar “recounting
measure”, analog of $\psi(x)$, but which requires to unravel the p-adic powers
grading of the zeta eigenvalues.

The “multiplicity” $a\log p=\log(p^{a})$ corresponds to the $a^{th}$ degree of
a p-adic number (Frobenius shift in $Z_{p}$).

We conjecture that $q$ must be relatively prime to $p$ (different), or
otherwise the distribution is again uniform.

######  Conjecture 5.2.

The distribution of zeta eigenvalues is unchanged under dilations by
$2\pi\alpha=\log n/q$ when $n$ is composite (e.g. diagram with $n=6$);

The other “trivial” case is conjectured to be when $p$ divides $q$, since we
believe that the zeta eigenvalues in the $p$-adelic sector are related to its
roots of unity, which correspond to divisors of $p-1=|Aut(F_{p},+)|$ [14]
333The symmetries of the tangent space to the abelian quantum group $Z_{p}$
[15]..

######  Conjecture 5.3.

The zeta eigenvalues “eliminated” by the dilation operator corresponding to
$2\pi\alpha/=\log p/q$, correspond to the roots of unity of $Z_{p}$, i.e. the
divisors of $p-1$.

This is supported by the number of minima and maxima of the resulting
distribution: $q|p-1$, which are exactly the descendents of $p$ in the POSet
of primes.

Before venturing into a recursive chase of roots of unity corresponding to the
Pratt tree of $p$ (symmetries of symmetries of symmetries … etc. [15]) we
should obtain more information on the above “leads”.

###  5.2. Additional examples and diagrams

Regarding the parameters $a$ and $q$ of the scaling operator with
$2\pi\alpha=a/q\cdot\log p$ it seems that the fundamental case is that
mentioned by Rademacher [8], when $a=1$ and $q=1$. A different $q$, like $q=3$
in Fig.1, has the effect to wrap the distribution $q$-times.

The real problem is, if there is a way to separate the zeros in families
associated to various primes, in a similar way the rational circle $Q/Z$
decomposes in the adelic duality.

##  6\. Primes versus Zeros: a Theoretical Exploration

The above ideas suggest a few questions to keep in mind : 1) Do the zeta
eigenvalues correspond to Hecke characters? (Teichmuler!?) 2) Are they local
integrals of Dirichlet characters? (Hecke finite order; p-adic?).

And what about a p-adic version of the distribution function $g_{\alpha}(t)$,
summing p-adic characters $exp(2\pi i\\{qt\\}_{p})$; does it localizes at some
of the zeroes themselves?

Indeed, for $2\pi\alpha=\log p$ the distribution can be rewritten as:

$g_{\alpha}(t)=\frac{\log p}{\pi}\frac{1}{1-p^{1/2}e^{2\pi it}}$

and seems to interpolate the zero eigenvalues (their projection on the
principal part). Modulo the normalization coefficient $\log/\pi$, it resembles
the value at $p$ of $L_{p}(s,\chi)$, an L-function associated to a Hecke
character $\chi$ [11], p.5. ($v=p,q_{v}=p$). Recall that any Hecke character
has the form $\omega_{s}=\omega\cdot|\ |^{s}$, with $\omega(x)=\chi(u),x=atu$
of finite order determined by a Dirichlet character [11], p.3. Then
$L(s,\omega)=L(0,\omega\omega_{s})$ (loc. cit. p.6). If $t_{v}(\omega)=p$ then
for $s=\rho$ (index of the zero omitted):

$L_{p}(\rho,\omega)=L_{p}(0,\omega_{\rho})=(1-t_{v}(\omega_{\rho}))q_{v}^{-\rho})^{-1}=\frac{1}{1-p^{1/2}e^{2\pi
i\gamma\log p/(2\pi))}}=\frac{1}{1-\sqrt{p}e^{2\pi i\lambda}}.$

Now $\omega_{\rho}$ is unramified and only depends on the coset $s+2\pi i/\log
pZ$ (see Eigendistributions [11], p.6). This seems to be compatible with the
form of the distribution $g_{\alpha}(t)$ of $\log p\cdot\gamma_{n}/(2\pi)$
i.e. suggesting $i\gamma_{n}\in 2\pi/\log p\in Z$ (or “close”) for a subfamily
of zero eigenvalues (some values $n$):

$s=\rho,\quad[s]=\frac{1}{2}+i\gamma+\frac{2\pi i}{\log
p}Z=\frac{1}{2}+\frac{2\pi i}{\log p}(\gamma\frac{\log p}{2\pi}+Z).$

A study in more depth the theory of $\omega$-eigendistributions, in view of
the relation with local zeta integrals, is mandatory; for example compare the
Mellin transform (local zeta integral) from [13], P 2.2.1, p.5 (author’s
notation: ${\mathcal{Z}}$ is the periodicization operator, MT is Mellin
transform and $f$ is the Gaussian distribution as a canonical choice of
Schwartz function) with formula (3.6) from [11] p.10, and the GCD
interpretation (Remark, p.11):

$\zeta(s)=\frac{MT({\mathcal{Z}}(GD))}{MT(f)}\quad\leftrightarrow\quad
L(s,\omega)=\frac{z(s,\omega)}{z_{0}(s,\omega)}=\frac{z(s,\omega;f)}{z_{0}(s,\omega),f>}.$

In fact, the local zeta integral for $p=\infty$ is Mellin transform:

$z(s,\omega_{s};f)=\int_{R_{+}}f(x)\omega_{s}(x)d\mu(x),\
\omega_{s}(x)=|x|^{s},d\mu(x)=dx/x\ (Haar\ measure),$

and the archimedean case example from [11], p.13, seems to the author to just
prove the above statement (Proposition 3.5). It also points to another
important property of the Gaussian distribution: it is a basis vector
(eigendistribution theory).

Since global zeta integrals have product form representations [11], p.17, we
speculate that the zeta function has a product decomposition in terms of the
L-functions corresponding to its local zeta integrals:

(4) $\zeta(s)=L_{\infty}(s,\omega_{\infty})\prod_{p}L_{p}(s,\omega_{p}).$

Here we can dispense of the need for analytic continuation either by
considering the alternating zeta function [26]:

$(1-2^{1-s})\zeta(s)=\sum_{n=1}^{\infty}(-1)^{n-1}n^{-s},$

or by interpreting the equality in the sense of distributions (zeta
integrals). The first factor is proportional to the $\Gamma$ function, and
responsible for the trivial zeros. Recall that

$L_{p}(\rho_{n},\omega_{p})=L_{p}(0,\omega_{\rho_{n}}\omega),\quad\omega_{\rho_{n}}\omega=\chi_{N(n)}|\
|^{\rho_{n}}.$

On the other hand (primes), the Riemann exact formula

$\psi(x)=x-\sum_{\rho}x^{\rho}/\rho-\sum_{-2n}x^{-2n}/(-2n)-\log(2\pi)$

corresponding to the logarithmic derivative of the zeta function, leads to the
distribution equation:

$\psi^{\prime}(x)=1-\sum_{\rho}x^{\rho}-\sum trivial\ zeros,$

while the LHS is a sum of Dirac’s delta distributions:

$\psi^{\prime}(x)=1-\sum_{p\in P,k\in N}\log p\ \delta_{p^{k}}.$

By analogy, the similar formula counting zeta zeros like in [3], interpreted
in the sense of distributions is formally similar to:

$\sum_{\rho}c_{\rho}\delta_{\rho}=\sum_{p}g_{p}(t)<->\sum_{p}\log
pL_{p}(\rho,\omega),$

since $g_{p}(t)$ seem to interpolate the scaled zeta eigenvalues. In fact the
analog formula should correspond, again, to the logarithmic derivative of the
zeta function as a product of L-functions: $\sum_{p}\log L_{p}(\rho,\omega)$.

Since we are more and more tentative in statements, a further study from the
point of view of theory of Hecke characters, the role of the Teichmuler p-adic
character and $\omega$-eigenvalues theory, is required at this stage.

##  7\. Conclusions

The primary goal of the presented investigations is to identify an
(pro-)algebraic structure underlying the Riemann spectrum, dual to the POSet
structure of primes:

$(R\mbox{-}Spec=\\{\theta_{n}\\}_{n}\in N,\ ?\
)\quad\leftrightarrow\quad(Z\mbox{-}Spec=\\{p_{n}\\},<<).$

In view of the Primes-Zeros duality [21], it is natural to expect that this
structure is of adelic origin.

Beyond the preliminary considerations presented above, more specific questions
need be addressed:

1) Is there a primary spectrum $\nu_{p}$ such that the Riemann spectrum [21]
is derived from, like in the case of a multiplicative function? Is this
structure dual to the POSet structure of the primes?

2) Does the finite-infinite prime dichotomy apply beyond the decomposition of
the zeta function $\zeta\cdot\Gamma-factor=\xi(s)$? We expect that the adelic
duality of $Q$ implies an Algebraic Quantum Group duality

$Q\to A_{alg}\to Hom(Q,T),$

where $T=\sum Q_{p}/Z_{p}$ is the “rational circle” (roots of unity), and that
this reflects in the behavior of the zeta eigenvalues under multiplication by
$\log p$ discovered by Rademacher, Ford and Zaharescu (the real scaling
invaries the distribution).

We refer to this idea graphically as:

$\textstyle{Primes:\ P\cup\infty}$$\textstyle{Zeros:\
\Theta+A}$$\textstyle{L-functions:\xi\quad 1/\Gamma}$$\textstyle{Adelic\
deccomposition:\theta_{n}=\sum_{p}\ \\{\ \\}_{p}+real}$

This adelic decomposition (of additive characters) could be related to the
partial fraction decomposition of the rationals, which in turn is related to
the Discrete Fourier Transform.

In conclusion, it seems that new ideas are needed for a break-through in the
case of RH. The general tendency is to move away from the analytic (NT)
approach initiated by Riemann, which seems to have reached its limitations,
and lean towards an adelic approach, notably the approach A. Connes is using
[25].

A direct investigation of the Riemann zeros and of their imaginary parts is
proposed, following [3], yet in connection with the POSet structure of prime
numbers. Since the zeros are dual to primes ($\delta_{p}$ basis, either in the
group algebra of the rationals as an algebraic quantum group, or as
distributions in the $\omega$-eigendistributions theory), there should be a
Fourier transform formula in terms of Hecke quasi-characters.

The work of Ford and Zaharescu is taken as a starting point, with emphasis on
the Rademacher case: the study of the change of the distribution of imaginary
parts of zeta zeros caused by the dilation operators
$L_{\alpha}(\gamma_{n})=\alpha\gamma_{n}$, when $2\pi\alpha=\log p$.

The interpretation of the distribution function $g_{\alpha}$ as a local
L-function is used to link it to the theory of $\omega$-eigendistributions for
such Hecke characters. The approach makes use of the work of many other
researchers providing various tools to the research table: S. Kudla, R. Meyer,
J. F. Burnol, A. Connes [11, 16, 17, 25] etc.

The theoretical understanding of Riemann Spectrum, in terms of p-adic
characters, $\omega$-eigendistributions etc, should be corroborated with a
computer investigation: 1) determine several distributions $g_{\log p/2\pi}$
using SAGE or Mathematica, and look for correlations between
$\lambda=\gamma_{n}/2\pi$ and primes $p$; 2) Investigate a p-adic
interpretation of $\theta_{n}$, “disguised” under their decimal form, as a
form of p-adic Haar wavelet analysis.

This proposed investigation was partially carried on, and reported in [27].

Returning to the theoretical investigation, since
$Z\mbox{-}Spec=Spec({\mathcal{A}}b)$, and the POSet structure is induced by
the loop “functor” $Aut_{Ab}$, the adelic duality categorified invites to
consider a Tannaka-Krein duality between ${\mathcal{A}}b_{f}$, the category of
finite abelian groups and some representation category, probably involving the
absolute Galois group, the rationals as an agebraic quantum group, and crystal
bases … a too big of a picture for the present author!

## Acknowledgments

I would like to thank G. Seelinger and P. Ding for their participation and
help in our weekly seminar on the Riemann Hypothesis; and for their patience
regarding the illusive and intangible outcomes.

## References

  * [1] V. Munoz and R. P. Marco, Unified treatment of explicit and trace formulas via Poisson-Newton formula, https://arxiv.org/abs/1309.1449
  * [2] O. Shanker, Entropy of Riemann zeta zero sequence, AMO - Advanced Modeling and Optimization, Volume 15, Number 2, 2013, https://camo.ici.ro/journal/vol15/v15b18.pdf
  * [3] K. Ford and A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, J. reine angew. Math. 579 (2005), 145-158. www.researchgate.net, 2005.
  * [4] L. M. Ionescu, A natural partial order on the prime numbers, Notes on Number Theory and Discrete Mathematics, Volume 21, 2015, Number 1, Pages 1?9; arxiv.org/abs/1407.6659, 2014.
  * [5] L.M. Ionescu, Remarks on physics as number theory,
http://www.gsjournal.net/old/files/4606_Ionescu2.pdf.

  * [6] V. R. Pratt, Every prime has a succinct certificate, SIAM J. Comput. Vol.4, No.3, Sept. 1975, 214-220.
  * [7] K. Ford, K. Soundararajan, A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, II, 0805.2745, 2009.
  * [8] H. Rademacher, Fourier analysis in number theory, Collected Works, pp.434-458.
  * [9] K. Conrad, The character group of $Q$,
http://www.math.uconn.edu/k̃conrad/blurbs/gradnumthy/characterQ.pdf

  * [10] I. M. Gel’fand M. I. Graev, I. I. Pyateskii-Shapiro, Representation theory and automorphic forms, Academic Press, 1990.
  * [11] S. S. Kudla, Tate’s thesis, An Introduction to the Langlands Program, pp 109-131, Editors: Joseph Bernstein, Stephen Gelbart, Springer, 2004.
  * [12] L. M. Ionescu, On primes numbers and Riemann hypothesis, NSF GP 2012, http://my.ilstu.edu/l̃miones/ISUP/ISU-GP.html
  * [13] P. Garrett, Riemann’s explicit/exact formula, 2015,
http://www-
users.math.umn.edu/g̃arrett/m/mfms/notes_2015-16/03_Riemann_and_zeta.pdf

  * [14] F. Gouvea, p-adic Numbers: An Introduction, Springer-Verlag, 1993.
  * [15] L. M. Ionescu, Recent presentations, http://my.ilstu.edu/ lmiones/presentations_drafts.htm
  * [16] R. Meyer, A spectral interpretation of the zeros of the Riemann zzeta function, math/0412277
  * [17] J. F. Burnol, Spectral analysis of the local conductor operator, math/9809119.pdf, 1998.
  * [18] John Baez, Quasicrystals and the Riemann Hypothesis,
golem.ph.utexas.edu/category/2013/06/quasicrystals_and_the_riemann.html

  * [19] Freeman Dyson, Frogs and Birds, AMS 56 (2009), 212-223.
  * [20] A. M. Odlyzko, Primes, quantum chaos and computers, Proc. Symp., May 1989, Washington DC, pp.35-46.
  * [21] B. Mazur, W. Stein, Primes: What is Riemann’s Hypothesis?, http://modular.math.washington.edu/rh/rh.pdf
  * [22] L. M. Ionescu, Categorification and group extensions, Applied Categorical Structures 10: 3547, 2002, Kluwer Academic Publishers; arxiv.org/abs/math/9906038
  * [23] L. M. Ionescu, A POSet structure on the set of primes, Number Theory Conference, UIUC, June 5-7, 2014, http://my.ilstu.edu/l̃miones/presentations_drafts.htm
  * [24] J-F. Burnol, The explicit formula and a propagator, http://arxiv.org/pdf/math/9809119.pdf
  * [25] A. Connes, An essay on the Riemann Hypothesis, http://www.alainconnes.org/docs/rhfinal.pdf
  * [26] Wikipedia: “Riemann Zeta Function”.
  * [27] L. M. Ionescu, A statistics study of Riemann zeros, 2014, to appear.