On the Connection Between Riemann Hypothesis and a Special Class of
  Neural Networks
The Riemann hypothesis (RH) is a long-standing open problem in mathematics.
It conjectures that non-trivial zeros of the zeta function all have real part
equal to 1/2. The extent of the consequences of RH is far-reaching and touches
a wide spectrum of topics including the distribution of prime numbers, the
growth of arithmetic functions, the growth of Euler totient, etc. In this note,
we revisit and extend an old analytic criterion of the RH known as the
Nyman-Beurling criterion which connects the RH to a minimization problem that
involves a special class of neural networks. This note is intended for an
audience unfamiliar with RH. A gentle introduction to RH is provided.
Soufiane Hayou
# On the Connection Between Riemann Hypothesis

###### Abstract

The Riemann hypothesis (\(\mathcal{RH}\)) is a long-standing open problem in mathematics. It conjectures that non-trivial zeros of the zeta function all lie on the line \(\text{Re}(z)=1/2\). The extent of the consequences of \(\mathcal{RH}\) is far-reaching and touches a wide spectrum of topics including the distribution of prime numbers, the growth of arithmetic functions, the growth of Euler's totient, etc. In this note, we revisit and extend an old analytic criterion of the \(\mathcal{RH}\) known as the Nyman-Beurling criterion which connects the \(\mathcal{RH}\) to a minimization problem that involves a special class of neural networks. This note is intended for an audience unfamiliar with \(\mathcal{RH}\). A gentle introduction to \(\mathcal{RH}\) is provided.

## 1 Introduction

The Riemann hypothesis conjectures that the non-trivial zeros of the Riemann zeta function are located on the line \(\text{Re}(z)=\frac{1}{2}\) in the complex plane \(\mathbb{C}\). This is a long-standing open problem in number theory first formulated by (Riemann, 1859). The Riemann zeta function was first defined for complex numbers \(z\) with a real part greater than \(1\) by \(\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^{z}},z\in\mathbb{C},\text{Re}(z)>1\). However, it is the extension of the zeta function \(\zeta\) to the whole complex plane \(\mathbb{C}\) that is considered in the statement of \(\mathcal{RH}\). This extension is called the _analytic continuation_ of the zeta function (details are provided in Appendix A).

There is strong empirical evidence that \(\mathcal{RH}\) holds. Recent numerical verification by Platt and Trudgian (2021) showed that \(\mathcal{RH}\) is at least true in the region \(\{z=a+ib\in\mathbb{C}:a\in(0,1),b\in(0,\gamma]\}\) where \(\gamma=3\cdot 10^{12}\), meaning that all zeros of the zeta function with imaginary parts in \((0,\gamma]\) have a real part equal to \(\frac{1}{2}\). Several other theoretical insights seem to support \(\mathcal{RH}\) ;we invite the reader to check Appendix A for a short summary of relevant results and insights. In this note, we are interested in an specific criterion of the \(\mathcal{RH}\), i.e. an equivalent statement of \(\mathcal{RH}\). This criterion is known as the Nyman-Beurling criterion (Nyman, 1950; Beurling, 1955) which states that \(\mathcal{RH}\) holds if and only if a special class of functions is dense in \(L_{2}(0,1)\). This class of functions can be seen as a special kind of neural networks with one dimensional input. In this note, we show that the sufficient condition can be easily extended to \(L_{2}((0,1)^{d})\). Specifically, we introduce a new class of neural networks and show that \(\mathcal{RH}\) implies the density of this class in \(L_{2}((0,1)^{d})\) for any \(d\geq 2\). The necessary condition in general dimension \(d\geq 2\) remains an open question.

## 2 Riemann Hypothesis

The Riemann zeta function was originally defined for complex numbers \(z\) with a real part greater than \(1\) by

\[\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^{z}},\quad z\in\mathbb{C},\text{Re}(z)>1. \tag{1}\]
The above definition of Riemann zeta function excludes the region of interest \(\{z\in\mathbb{C}:\text{Re}(z)=\frac{1}{2}\}\) since the series in Eq. (3) diverge when \(|z|<1\). Indeed, \(\mathcal{RH}\) is stated for the an extension of the zeta function on the whole complex plane \(\mathbb{C}\). This extension is called the analytic continuation, and it is unique by the Identity theorem (Walz, 2017). To give the reader some intuition of how such extension is defined, let us show how we can extend \(\zeta\) to the region \(\{z\in\mathbb{C}:\text{Re}(z)>0\}\). Observe that the function \(\zeta\) satisfies the following identity

\[(1-2^{1-z})\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^{z}}-2\sum_{n=1}^{\infty} \frac{1}{(2n)^{z}}=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{z}},\]

where the right hand side is defined for any complex number \(z\) such that \(\text{Re}(z)>0\). Using similar techniques, we can show that for any \(z\in\mathbb{C}\) such that \(\text{Re}(z)\in(0,1)\),

\[\zeta(z)=2^{z}\pi^{z-1}\sin\left(\frac{\pi z}{2}\right)\Gamma(1-z)\zeta(1-z), \tag{2}\]

which helps extend \(\zeta\) to complex numbers with negative real part. A step by step explanation of the analytic continuation of the \(\zeta\) function is provided in Appendix A.

Zeros of the \(\zeta\) function.From Eq. (2), we have \(\zeta(-2k)=0\) for any integer \(k\geq 1\). The negative even integers \(\{-2k\}_{k\geq 1}\) are thus called _trivial zeros_ of the Riemann zeta function since the result follows from the simple fact that \(\sin\left(-\pi k\right)=0\) for all integers \(k\geq 1\). The other zeros of \(\zeta\) are called non-trivial zeros, and their properties remain poorly understood. The \(\mathcal{RH}\) conjectures that they all lie on a the line \(\text{Re}(z)=\frac{1}{2}\).

Riemann Hypothesis (\(\mathcal{RH}\))._All non-trivial zeros of \(\zeta\) have a real part equal to \(\frac{1}{2}\)._

Whether \(\mathcal{RH}\) holds is still an open question. The consequences of the Riemann hypothesis are various (see Appendix A) and numerous equivalent results exist in the literature. In the next section, we re-visit an old analytic criterion of \(\mathcal{RH}\)that involves a special type of functions that can be seen as single layer neural networks.

### A _Neural Network_ Criterion for \(\mathcal{RH}\)

For \(p>1,d\in\mathbb{N}\backslash\{0\}\), and some set \(S\subset\mathbb{R}^{d}\), let \(L_{p}(S)\) denote the set of real-valued functions \(f\) defined on \(S\) such as \(|f|^{p}\) is Lebesgue integrable, i.e. \(L_{p}(S)=\{f:S\to\mathbb{R}:\int_{S}|f|^{p}d\mu<\infty\},\) where \(\mu\) is the Lebesgue measure on \(\mathbb{R}^{d}\). We denote by \(\|.\|_{p}\) the standard Lebesgue norm defined by \(\|f\|_{p}=\left(\int_{S}|f|^{p}d\mu\right)^{1/p}\) for \(f\in L_{p}(S)\).

For some \(k\geq 1\), let \(I_{k}\stackrel{{ def}}{{=}}(0,1)^{k}=(0,1)\times\cdots\times(0,1)\) where the product contains \(k\) terms. Let \(\rho\) denote the fractional part function given by \(\rho(x)=x-\lfloor x\rfloor,\) for \(x\in\mathbb{R}\). Consider the following class of functions defined on the interval \(I_{1}\)

\[\mathcal{N}=\{f(x)=\sum_{i=1}^{m}c_{i}\rho\left(\frac{\beta_{i}}{x}\right),x \in I_{1}:m\geq 1,c\in\mathbb{R}^{m},\beta\in I_{m},c^{T}\beta=0\}.\]

In machine learning nomenclature, \(\mathcal{N}\) consists of single-layer neural networks with a constrained parameter space and a specific non-linearity (or activation function) that depends on the fractional part \(\rho\). The parameters \((c,\beta)\) belong to the set \(\{c\in\mathbb{R}^{m},\beta\in(0,1)^{m},c^{T}\beta=0\}\). The values \((\rho(\beta_{i}/x))_{1\leq i\leq m}\) act as the neurons (post-activations) in the neural network. In Fig. 1, we depict
neuron values for different choices of \(\beta_{i}\). The graphs show fluctuations when \(x\) is close to \(0\) which should be expected since the function \(x\to\rho(\beta_{i}/x)\) fluctuates indefinitely between \(0\) and \(1\) as \(x\) goes to zero, whenever \(\beta_{i}\neq 0\). In figure Fig. 1 (right), we show an example of a function from the class \(\mathcal{N}\) given by \(f(x)=\rho(0.7/x)-\rho(0.3/x)-4\rho(0.1/x)\). We observe that \(f\) is a step function which might be surprising at first glance. However, it is easy to see that \(\mathcal{N}\) consists only of step functions. This is due to the constraint on the parameters \(c,\beta\), and the fact that \(\rho(x)=x-\lfloor x\rfloor\). Now, we are ready to state the main results that draw an interesting connection between \(\mathcal{RH}\) and the class \(\mathcal{N}\).

**Theorem 1** (Nyman (1950)): _The \(\mathcal{RH}\) is true if and only if \(\mathcal{N}\) is dense in \(L_{2}(I_{1})\)._

Beurling (1955) later extended this result by showing that for any \(p>1\), the \(\zeta\) function has no zeroes in the set \(\{z\in\mathbb{C}:\text{Re}(z)>1/p\}\) if and only if the set \(\mathcal{N}\) is dense in \(L_{p}(I_{1})\).

**Theorem 2** (Beurling (1955)): _The Riemann zeta function is free from zeros in the half plane \(Re(z)>\frac{1}{p},1<p<\infty\), if and only if \(\mathcal{N}\) is dense in \(L_{p}(I_{1})\)._

The intuition behind this connection is rather simple. The number of fluctuations of the function \(x\to\rho(\beta/x)\) near \(0\) is closely related to the \(\zeta\) function. To understand the machinery of the proofs of Theorem 1 and Theorem 2, we provide a sketch of the proof by Beurling (1955) for the sufficient condition in Appendix B. Using the same techniques, we derive the following result on zero-free regions of the zeta function.

**Lemma 1** (Nyman-Beurling zero-free regions): _Let \(f\in\mathcal{N}\) and \(\delta=\|1-f\|_{2}\) be the distance between the constant function \(1\) on \(I_{1}\) and \(f\). Then, the region \(\{z\in\mathbb{C},\text{Re}(z)>\frac{1}{2}\left(1+\delta^{2}|z|^{2}\right)\}\) is free of zeroes of the Riemann zeta function \(\zeta\)._

The condition that \(\mathcal{N}\) should be dense in \(L_{2}(I_{1})\) can be replaced by the following weaker condition: the constant function \(1\) on \(I_{1}\) can be approximated up to an arbitrary accuracy with functions from \(\mathcal{N}\). This is because from the constant function \(1\), one can construct an approximation of any step-wise function, which in turn can approximate any function in \(L_{2}(I_{1})\).

A discussion on the empirical implications of Theorem 1 is provided in Appendix B. In the next section, we show that the sufficient condition of Theorem 2 can be easily generalized to networks with multi-dimensional inputs, i.e. the case \(d\geq 1\).
## 3 A sufficient condition in the multi-dimensional case

Let \(d\geq 1\) and consider the following class of neural networks with inputs in \(I_{d}\),

\[\mathcal{N}_{d}=\{f(x)=\sum_{j=1}^{d}\sum_{i=1}^{m}c_{i,j}\rho\left(\frac{\beta_{ i,j}}{x_{j}}\right),x\in I_{d}:m\geq 1,c\in\mathbb{R}^{d\times m},\beta\in I_{d \times m},c^{T}\beta=0\},\]

where \(c=(c_{1,1},c_{2,1},\ldots,c_{m,1},\ldots,c_{m,d})^{\top}\in\mathbb{R}^{md}\) is the flattened vector of \((c_{\cdot,j})_{1\leq j\leq d}\). Notice that we recover the Nyman-Beurling class \(\mathcal{N}\) when \(d=1\). Using this class, we can generalize the zero-free region result given by Theorem 1 to a multi-dimensional setting in the case \(p=2\).1

Footnote 1: The choice of \(p=2\) is arbitray, and similar result to that of Theorem 2 can be obtained for any \(p>1\).

**Lemma 2** (zero-free regions for general \(d\geq 1\)): _Let \(d\geq 1\) and \(f\in\mathcal{N}_{d}\). Let \(\delta=\|1-f\|_{2}\) be the \(L_{2}\) distance between the constant function \(1\) on \(I_{d}\) and \(f\). Then, the region \(\{z\in\mathbb{C},\mathrm{Re}(z)>\frac{1}{2}\left(1+\delta^{\frac{d}{d}}|z|^{2} \right)\}\) is free of zeroes of the Riemann zeta function \(\zeta\)._

In Fig. 2, we depict the zero-free regions from Lemma 2. The smaller the constant \(\delta\), the larger the region. The multi-dimensional input case (\(d\geq 2\)) can therefore be interesting if we can better approximate the constant function \(1\) with functions from \(\mathcal{N}_{d}\). More precisely, the result of Lemma 2 is relevant if for some \(d\geq 2\), we could find \(\delta\) such that \(\delta^{2/d}<\delta_{1}\), where \(\delta_{1}\) is the approximation error in the one-dimensional case \(d=1\). In this case, the zero-free region obtained with \(d\geq 2\) will be larger than the one obtained with \(d=1\). We refer the reader to Section 4 for a more-in depth discussion about the empirical implications of the multi-dimensional case. Notice that if \(\delta\) can be chosen arbitrarily small, then the zero-free region in Lemma 2 can be extended to the whole half-plane \(\{\mathrm{Re}(z)>1/2\}\). This is a generalization of the sufficient condition of Theorem 2 in the multi-dimensional case.

**Corollary 3** (Sufficient condition for \(d\geq 1\)): _Let \(d\geq 1\). Assume that the class \(\mathcal{N}_{d}\) is dense in \(L_{2}(I_{d})\). Then, the region \(\{\mathrm{Re}(z)>1/2\}\) is free of the zeroes of the Riemann zeta function \(\zeta\)._

### Open problem: The necessary condition for \(d\geq 2\)

By considering the class \(\mathcal{N}_{d}\), we generalized the sufficient condition of Beurling's criterion in the multi-dimensional input case \(d\geq 2\). However, it is unclear whether a similar necessary condition holds. Proving that \(\mathcal{RH}\) implies the density of \(\mathcal{N}_{d}\) in \(L_{2}(I_{d})\) is challenging. A function \(f\in\mathcal{N}_{d}\) can be expressed as \(f(x)=\sum_{i=1}^{d}f_{i}(x_{i})\) for \(x=(x_{1},\ldots,x_{d})^{\top}\in I_{d}\), and \(f_{i}\) are functions with one-dimensional inputs. This special additive form of functions from \(\mathcal{N}_{d}\) makes it harder to use arguments similar to the one-dimensional case (Theorem 2) to prove density results.

Figure 2: Zero-free regions of the form \(\{\mathrm{Re}(z)>\frac{1}{2}(1+\Delta|z|^{2})\}\) as stated in Theorem 1, Theorem 2, and Theorem 4.
## 4 Discussion on the Implications and Limitations

In this section, we discuss some empirical implications of Lemma 1 and Lemma 2.

Probabilistic zero-free regions.Notice that Lemmas 1 and 2 require access to the distance \(\|1-f\|_{2}\) which is generally intractable. However, we can approximate this quantity using Monte Carlo samples and obtain high probability bounds for this norm. Hence, the best we can do with such criterion is to verify the non-existence of zeroes of \(\zeta\) in some region _with high probability_. Indeed, using Hoeffding's inequality, we have the following result.

**Lemma 4**: _Let \(d\geq 1\), \(N\geq 1\) and \(X_{1},X_{2},\ldots,X_{N}\) be iid uniform random variables on \(I_{d}\). Let \(f\in\mathcal{N}_{d}\) (where for \(d=1\), we denote \(\mathcal{N}_{d}=\mathcal{N}\)) such that \(f(x)=\sum_{j=1}^{d}\sum_{i=1}^{m}c_{i,j}\rho\left(\frac{\beta_{i,j}}{x_{j}}\right)\) for all \(x\in I_{d}\), for some \(m\geq 1,\beta\in I_{m\times d},c\in\mathbb{R}^{m\times d}\). Then, for any \(\alpha\in(0,1)\), we have with probability at least \(1-\alpha\), the region \(R_{N}\stackrel{{\text{def}}}{{=}}\{\text{Re}(z)>\frac{1}{2}\left( 1+\Delta_{N}(f)^{1/d}|z|^{2}\right)\}\) is free of the zeroes of \(\zeta\), where \(\Delta_{N}(f)=\frac{1}{N}\sum_{i=1}^{N}(1-f(x_{i}))^{2}+(1+\|c\|_{1}^{2})\sqrt {\frac{2\log(2/\alpha)}{N}}\), with \(\|c\|_{1}=\sum_{i=1}^{m}|c_{i}|\)._

The proof follows from a simple application of Hoeffding's concentration inequality to control the deviations of the empirical risk \(N^{-1}\sum_{i=1}^{N}(1-f(x_{i}))^{2}\). Hoeffding's lemma requires that the random variables \((1-f(x_{i}))^{2}\) are bounded, which is straightforward since \((1-f(x_{i}))^{2}\leq 2(1+f(x_{i})^{2})\leq 2(1+\|c\|_{1}^{2})\) almost surely.

The result of Theorem 4 has an important implication on the choice of the sample size. Indeed, to have the coefficient \(\delta_{N}^{1/d}\) of order \(\epsilon\) with high probability, a necessary condition is that \(N=\mathcal{O}(\epsilon^{2d})\).

When is the multi-dimensional variant better than the one-dimensional criterion?For some \(d\geq 2\), it is straightforward that the multi-dimensional criterion given in Lemma 2 is better than the one given in Lemma 1 only if \(\inf_{f\in\mathcal{N}_{d}}\|1-f\|_{2}^{1/d}<\inf_{f\in\mathcal{N}}\|1-f\|_{2}\). Under this condition, the zero-free region is larger with \(d\geq 2\). For empirical verification of the \(\mathcal{RH}\) and for same probability threshold \(\alpha\), Lemma 4 implies that the multi-dimensional setting is better than the one-dimensional counterpart whenever \(\inf_{f\in\mathcal{N}_{d}}\Delta_{N}(f)^{1/d}<\inf_{f\in\mathcal{N}}\Delta_{N}(f)\). We discuss the feasibility of such conditions in the next paragraph.

What does it take to improve upon existing numerical verifications of \(\mathcal{RH}\)?The high probability zero-free regions from Lemma 4 are of the form \(\{\text{Re}(z)>\frac{1}{2}(1+\Delta|z|^{2})\}\) for some constant \(\Delta>0\).

Using a different analytical criterion of the \(\mathcal{RH}\), Platt and Trudgian (2021) showed that the region \(\{a+ib:a\in(0,1),b\in(0,\gamma],\gamma\approx 3\cdot 10^{12}\}\) is free of the zeroes of \(\zeta\). Hence, using Lemma 4 to improve this result requires that the region \(R_{N}\cap\{a+ib,a\in(0,1),b\in(0,\gamma]\}\) contains complex numbers \(z\) with imaginary part larger than order \(10^{12}\). Let \(z=a+ib\in\mathbb{C}\). Having \(z\in R_{N}\) implies that \(b^{2}<-a^{2}+\Delta_{N}(f)^{-1/d}(2a-1)\). For the region of interest where \(a\in(0,1)\), and assuming that \(\Delta_{N}(f)\) is small enough, the right-hand side is of order \(\Delta_{N}(f)^{-1/d}(2a-1)\) which is maximized for \(a=1\) and equal to \(\Delta_{N}(f)^{-1/d}\). Thus, to improve upon existing work (Platt and Trudgian, 2021) (at least with some high probability certificate), we need to have \(\Delta_{N}(f)^{-1/d}\) of order \(10^{12}\), which means that \(\Delta_{N}(f)^{1/d}\) should be at least of order \(10^{-12}\). This requires a the minimize of a the empirical risk \(N^{-1}\sum_{i=1}^{N}(1-f(x_{i}))^{2}\) with a minimum sample size of order \(10^{24}\) which is unfeasible with the current compute resources.