appendixd.tex
\centerline{\bf Appendix D:  Tabulated numerical values}
%\begin{table}[h]\centering
\begin{tabular}{|c|c|c|}   
\hline
$k$ & $\lambda_k$ & $\eta_k$ \\
\hline
0 &   $ ~ $     & -0.577216  \\   
1 & 0.0230957   & 0.187546   \\
2 & 0.0923457   & -0.0516886 \\
3 & 0.207639    & 0.0147517 \\
4 & 0.368793    & -0.00452448\\
5 & 0.575543    & 0.0014468 \\
6 & 0.827566    & -0.000471544 \\
7 & 1.12446     & 0.00015518 \\
8 & 1.46576     & -0.0000513452 \\
9 & 1.85092     & 0.0000170414 \\
10 & 2.27934    & $-5.66605 \times 10^{-6}$ \\
11 & 2.75036    & $ 1.88585 \times 10^{-6}$ \\
12 & 3.26326    & $ -6.28055 \times 10^{-7}$ \\ 
13 & 3.81724    & $ 2.09241 \times 10^{-7}$ \\
14 & 4.41148    & $ -6.97247 \times 10^{-8}$ \\
15 & 5.04508    & $ 2.32372 \times 10^{-8}$ \\
16 & 5.71711    & $ -7.74484 \times 10^{-9}$ \\
17 & 6.42658    & $ 2.58144 \times 10^{-9}$ \\
18 & 7.17248    & $ -8.60444 \times 10^{-10}$ \\
19 & 7.95374    & $ 2.86808 \times 10^{-10}$ \\
20 & 8.76928    & $ -9.56012 \times 10^{-11}$ \\
21 & 9.61796    & $ 3.18668 \times 10^{-11}$ \\
22 & 10.4986    & $ -1.06222 \times 10^{-11}$ \\
23 & 11.4101    & $ 3.54072 \times 10^{-12}$ \\
24 & 12.3513    & $ -1.18024 \times 10^{-12}$ \\
25 & 13.3210    & $ 3.93412 \times 10^{-13}$ \\
26 & 14.3179    & $ -1.31137 \times 10^{-13}$ \\
27 & 15.3408    & $ 4.37124 \times 10^{-14}$ \\
28 & 16.3885    & $ -1.45708 \times 10^{-14}$ \\
29 & 17.4599    & $ 4.85694 \times 10^{-15}$ \\
30 & 18.5538    & $ -1.61898 \times 10^{-15}$ \\
\hline
\end{tabular}
%\end{table}

%\begin{table}[h]\centering
\begin{tabular}{|c|c|c|}  
\hline
$k$ & $\lambda_k$ & $\eta_k$ \\
\hline
31 & 19.6689    & $ 5.3966 \times 10^{-16}$ \\
32 & 20.8041    & $ -1.79887 \times 10^{-16}$ \\
33 & 21.9582    & $ 5.99622 \times 10^{-17}$ \\
34 & 23.1301    & $ -1.99874 \times 10^{-17}$ \\
35 & 24.3188    & $ 6.66247 \times 10^{-18}$ \\
36 & 25.5232    & $ -2.22082 \times 10^{-18}$ \\
37 & 26.7422    & $ 7.40274 \times 10^{-19}$ \\
38 & 27.9749    & $ -2.46755 \times 10^{-19}$ \\
39 & 29.2202    & $ 8.22527 \times 10^{-20}$ \\
40 & 30.4774    & $ -2.74176 \times 10^{-20}$ \\
41 & 31.7454    & $ 9.13919 \times 10^{-21}$ \\
42 & 33.0236    & $ -3.0464 \times 10^{-21}$ \\
43 & 34.3111    & $ 1.01547 \times 10^{-21}$ \\
44 & 35.6072    & $ -3.38488 \times 10^{-22}$ \\
45 & 36.9113    & $ 1.12829 \times 10^{-22}$ \\
46 & 38.2227    & $-3.76098 \times 10^{-23}$ \\
47 & 39.5408    & $ 1.25366 \times 10^{-23}$ \\
48 & 40.8653    & $ -4.17887 \times 10^{-24}$ \\
49 & 42.1955    & $ 1.39295 \times 10^{-24}$ \\
50 & 43.5311    & $ -4.64318 \times 10^{-25}$ \\
51 & 44.8718    & $ 1.54773 \times 10^{-25}$ \\
52 & 46.2172    & $ -5.15909 \times 10^{-26}$ \\
53 & 47.5671    & $ 1.71970 \times 10^{-26}$ \\
54 & 48.9214    & $ -5.73232 \times 10^{-27}$ \\
55 & 50.2798    & $ 1.91077 \times 10^{-27}$ \\
56 & 51.6423    & $ -6.36925 \times 10^{-28}$ \\
57 & 53.0089    & $ 2.12308 \times 10^{-28}$ \\
58 & 54.3795    & $ -7.07695 \times 10^{-29}$ \\
59 & 55.7542    & $ 2.35898 \times 10^{-29}$ \\
60 & 57.1331    & $ -7.86327 \times 10^{-30}$ \\
\hline
\end{tabular}

\begin{tabular}{|c|c|c|}  
\hline
$k$ & $\lambda_k$ & $\eta_k$ \\
\hline
61 & 58.5163    & $ 2.62109 \times 10^{-30}$ \\
62 & 59.9039    & $ -8.73697 \times 10^{-31}$ \\
63 & 61.2962    & $ 2.91232 \times 10^{-31}$ \\
64 & 62.6934    & $ -9.70775 \times 10^{-32}$ \\
65 & 64.0957    & $ 3.23591  \times 10^{-32}$ \\ 
66 & 65.5033    & $ -1.07864 \times 10^{-32}$ \\
67 & 66.9167    & $  3.59546 \times 10^{-33}$ \\
68 & 68.3361    & $ -1.19849 \times 10^{-33}$ \\
69 & 69.7618    & $  3.99496 \times 10^{-34}$ \\
70 & 71.1942    & $ -1.33165 \times 10^{-34}$ \\
71 & 72.6337    & $ 4.43884 \times 10^{-35}$ \\
72 & 74.0805    & $ -1.47961 \times 10^{-35}$ \\
73 & 75.5350    & $ 4.93205 \times 10^{-36}$ \\
74 & 76.9976    & $ -1.64402 \times 10^{-36}$ \\
75 & 78.4686    & $ 5.48005 \times 10^{-37}$ \\
76 & 79.9484    & $ -1.82668 \times 10^{-37}$ \\
77 & 81.4373    & $ 6.08895 \times 10^{-38}$ \\
78 & 82.9357    & $ -2.02964 \times 10^{-38}$ \\
79 & 84.4437    & $ 6.76550 \times 10^{-39}$ \\
80 & 85.9617    & $ -2.25516 \times 10^{-39}$ \\
81 & 87.4900    & $ 7.51722 \times 10^{-40}$ \\
82 & 89.0288    & $ -2.50574 \times 10^{-40}$ \\
83 & 90.5782    & $ 8.35246 \times 10^{-41}$ \\
84 & 92.1386    & $ -2.78415 \times 10^{-41}$ \\
85 & 93.7099    & $ 9.28051 \times 10^{-42}$ \\
86 & 95.2924    & $ -3.09350 \times 10^{-42}$ \\
87 & 96.8862    & $ 1.03117 \times 10^{-42}$ \\
88 & 98.4912    & $ -3.43723 \times 10^{-43}$ \\
89 & 100.1076   & $ 1.14574 \times 10^{-43}$ \\
90 & 101.7352   & $ -3.81914 \times 10^{-44}$ \\
\hline
\end{tabular}

\begin{tabular}{|c|c|c|}  
\hline
$k$ & $\lambda_k$ & $\eta_k$ \\
\hline
91 & 103.3741   & $ 1.27304 \times 10^{-44}$ \\
92 & 105.0242   & $ -4.24349 \times 10^{-45}$ \\
93 & 106.6852   & $ 1.41450 \times 10^{-45}$ \\
94 & 108.3572   & $ -4.71500 \times 10^{-46}$ \\
95 & 110.0398   & $ 1.57166 \times 10^{-46}$ \\
96 & 111.7328   & $ -5.23888 \times 10^{-47}$ \\
97 & 113.4361   & $ 1.74629 \times 10^{-47}$ \\
98 & 115.1492   & $ -5.82098 \times 10^{-48}$ \\
99 & 116.8719   & $ 1.94032 \times 10^{-48}$ \\
100 & 118.6038  & $ -6.46775 \times 10^{-49}$ \\
\hline
\end{tabular}
%\end{table}
appendixe.tex
\centerline{\bf Appendix EI:  Sum estimations and lower bounds pertinent 
to other Dirichlet functions}

\medskip
Analogous to the Corollary which we have presented below Eq. (24) of the
text, here we develop similar lower bounds appropriate for explicit
formulas for Dirichlet and Hecke L-functions.  We make substantial use of
the very recent results of Ref. \cite{li2}, of which we need to recall 
some details.  We relegate to the end of this first part of the Appendix 
some relations concerning elementary sums.  In the second part, we provide
independent derivations of the major results, Theorems 1 and 2, of Ref.
\cite{li2}.

Let $\chi$ be a primitive Dirichlet character of modulus $r$, and 
$L(s,\chi)$ the Dirichlet L-function of character $\chi$.  The function
$$\xi(s,\chi)=\left({\pi \over r}\right )^{-(s+a)/2}\Gamma\left (
{{s+a} \over 2}\right )L(s,\chi), \eqno(E.1)$$
where
$a$ is $0$ if $\chi(-1)=1$ and $a$ is $1$ if $\chi(-1)=-1$, 
satisfies the functional equation $\xi(s,\chi)=\epsilon_\chi\xi(1-s,\bar{\chi})$,
with $\epsilon_\chi$ a constant of absolute value one.  The function
$\xi(s,\chi)$ is an entire function of order one and has a product 
representation $\xi(s,\chi)=\xi(0,\chi)\prod_\rho(1-s/\rho)$, where the
product is over all the zeros of $\xi(s,\chi)$.

We put
$$\lambda_\chi(n)=\sum_\rho\left[1-\left(1-{1 \over \rho}\right)^n\right ],
~~~~~~~~n \geq 1. \eqno(E.2)$$
We presume that $\lambda_\chi(n) > 0$ for all $n=1,2,\ldots$ if and only if 
all of the zeros of $\xi(s,\chi)$ are located on the critical line Re $s=1/2$.
%this last statement needs to be verified with Professor Li ................
Then, Li has obtained \cite{li2}
$$\lambda_\chi(n)=S_\chi(n)+{n \over 2}\left(\ln{r \over \pi}-\gamma\right)
+\tau_\chi(n), \eqno(E.3)$$
where
$$S_\chi(n) \equiv -\sum_{j=1}^n{n \choose j}{{(-1)^{j-1}} \over {(j-1)!}}
\sum_{k=1}^\infty {{\Lambda(k)} \over k} \bar{\chi}(k)(\ln k)^{j-1},$$
$$=-\sum_{k=1}^\infty {{\Lambda(k)} \over k} \bar{\chi}(k)L_{n-1}^1(\ln k),
\eqno(E.4)$$
$$\tau_\chi(n)=\sum_{j=2}^n{n \choose j}(-1)^j(1-2^{-j})\zeta(j)-{n \over 2}
\sum_{\ell=1}^\infty {1 \over {\ell(2\ell-1)}} ~~~~\mbox{for}~~~~ \chi(-1)=1,$$
$$= \sum_{j=2}^n{n \choose j}(-1)^j2^{-j}\zeta(j) ~~~~~~~~~~~~~~~~\mbox{for}~~~~ 
\chi(-1)=-1,  \eqno(E.5)$$
and $L_n^\alpha$ is an associated Laguerre polynomial.
We recall Eq. (25) of the text,
$$\tau_\chi(n)=S_1 - n\ln 2 ~~~~~~~~~~~~~~~~~~~\mbox{if} ~~~~\chi(-1)=1.$$
$$~~~~~~~~=S_0-S_1, ~~~~~~~~~~~~~~~~~~~   \mbox{if} ~~~~\chi(-1)=-1.  
\eqno(E.6)$$
where $S_0$ is defined in Eq. (A.15) of Appendix A.  Therefore, by using the
summation estimations presented in Appendix A, we obtain
$$\lambda_\chi \geq S_\chi(n) + {n \over 2}\ln n+{n \over 2}\left(\ln {r 
\over \pi}-1-2\ln2 \right)+{1 \over 2} ~~~~~\mbox{for} ~~~~\chi(-1)=1,$$
$$\geq S_\chi(n) + {n \over 2}\ln n+{n \over 2}\left(\ln {r \over \pi} - 1
\right) + {1 \over 2} ~~~~\mbox{for} ~~~~\chi(-1)=-1.  \eqno(E.7)$$
In accord with the discussion of the text, we conjecture that the sum
$S_\chi$ is 'small'.  By this we mean that $S_\chi(n)$ could be $O(n)$ and 
probably even $S_\chi$ is $O(n^{1/2+\epsilon})$, for $\epsilon >0$.  %not $O(\ln n)$
This could result from a near 
exponential amount of cancellation in this sum due to the phases present in 
the Dirichlet characters.

We next introduce the function
$$\xi_E(s)=c_EN^{s/2}(2\pi)^{-s}\Gamma\left(s+{1 \over 2}\right )L_E\left(s + 
{1 \over 2}\right), \eqno(E.8)$$
where $L_E$ is the L-series associated with an elliptic curve $E$ over the 
rational numbers, $N$ is the conductor, and $c_E$ is a constant chosen so 
that $\xi_E(1)=1$ \cite{bump,mestre}.
The function of Eq. (E.8) is an entire function of order one and satisfies 
$\xi_E(s) =w\xi_E(1-s)$ where $w=(-1)^r$ with $r$ being the vanishing order 
of $\xi_E(s)$ at $s=1/2$.

We let 
$$\lambda_E(n) = \sum_\rho \left[1-\left(1-{1 \over \rho}\right)^n \right ],
~~~~~~~~n \geq 1, \eqno(E.9)$$
where the sum is over all zeros $\rho$ of $\xi_E(s)$.  All of these zeros lie 
on the critical line if and only if \cite{li2} $\lambda_E(n) > 0$ for all 
$n=1,2,\ldots$.

Now Li \cite{li2} has obtained the explicit formula
$$\lambda_E(n) = S_E +n\left (\ln {\sqrt{N} \over {2\pi}}-\gamma \right )
+n\left(-{2 \over 3}+\sum_{\ell=1}^\infty {3 \over {\ell(2\ell+3)}}\right )
+\sum_{j=2}^n {n \choose j}(-1)^j \sum_{\ell=1}^\infty {1 \over {(\ell+1/2)^j}}, 
\eqno(E.10)$$
where
$$S_E(n) \equiv -\sum_{j=1}^n{n \choose j}{{(-1)^{j-1}} \over {(j-1)!}}
\sum_{k=1}^\infty {{\Lambda(k)} \over k^{3/2}} b(k) (\ln k)^{j-1},$$
$$=-\sum_{k=1}^\infty {{\Lambda(k)} \over k^{3/2}} b(k) L_{n-1}^1(\ln k).
\eqno(E.11)$$
In Eq. (E.11), $b(p^k)=a_p^k$ if $p|N$ and $b(p^k)=\alpha_p^k + \beta_p^k$ if
$(p,N)=1$, where for each prime number $p$, $\alpha_p$ and $\beta_p$ are the 
roots of the equation $T^2-a_pT+p$ and the values of $a_p$ are connected with 
the reduction of $E$ at $p$ \cite{li2}.

We recall Eq. (26) of the text, so that we may write
$$\lambda_E(n) = S_E(n) +n\left (\ln {\sqrt{N} \over {2\pi}}-\gamma \right )
+2\left ( 1 - {1 \over 3}\ln 2 \right )n + S_3 - S_0 + 2n + (-1)^n - 1,  
\eqno(E.12)$$
where the zeta function sum $S_3$ is defined in Eq. (A.18) of Appendix A (and 
see below, Eq. (E.15)).  Then, by the results of Appendix A we obtain
$$\lambda_E(n) \geq S_E(n) + n \ln n + n \ln {\sqrt{N} \over {2\pi}} + \left 
(3 + {4 \over 3}\ln 2 \right)n - 1. \eqno(E.13)$$
Again, we conjecture that the sum $S_E$ is $O(n^{1/2+\epsilon})$.  The values of 
$a_p$ include $0$ and $\pm 1$, so that the values of $b(p^k)$ can either be zero or 
include significant sign alternation when $p|N$.  Similarly, for $(p,N)=1$, 
the roots of $T^2-a_p T+p$ can include $\pm \sqrt{p}$ and $[\pm 1 \pm 
\sqrt{1-4p}]/2$, giving various sign changes in $b(p^k)$.     
%want to figure out the range of values of a_p when E has good reduction at p,
%see p. 4 of Li's preprint .......................................................
When $E$ has good reduction at $p$, $-2\sqrt{p} \leq a_p \leq 2\sqrt{p}$, so 
that it again appears that $b(p^k)$ can have significant changes in sign, 
possibly leading to much cancellation in $S_E$.

Concerning Eqs. (E.6) and (E.12) we record and briefly discuss some 
elementary summation results.  We have
$$\ln 2 = \sum_{n=1}^\infty {1 \over {2n-1}} - {1 \over 2}\sum_{n=1}^\infty 
{1 \over n}, \eqno(E.14a)$$
giving
$$\ln 2 = {4 \over 3} - 3\sum_{n=1}^\infty {1 \over {2n(2n+3)}}, 
\eqno(E.14b)$$
leading to 
$$-{2 \over 3} + \sum_{n=1}^\infty {3 \over {n(2n+3)}} = 2(1 - \ln 2).  
\eqno(E.14c)$$
In addition, we have
$$\sum_{\ell=1}^\infty {1 \over {(\ell+1/2)^j}} = \sum_{m=1}^\infty {2^j 
\over {(2m+1)^j}} =\sum_{m=1}^\infty \left ({2 \over {2m+1}}\right )^j + 
\sum_{m=1}^\infty \left ({2 \over {2m}}\right )^j - \zeta(j)$$
$$=\sum_{k=3, \mbox{odd}}^\infty \left ({2 \over k}\right )^j + 
\sum_{k=2, \mbox{even}}^\infty \left({2 \over k}\right )^j -\zeta(j)$$
$$=\sum_{k=2}^\infty \left ({2 \over k}\right)^j - \zeta(j) = 2^j[\zeta(j) - 1] 
- \zeta(j).  \eqno(E.15)$$
This equation is a restatement of the relation between the Riemann zeta 
function and the Hurwitz zeta function $\zeta(s,a)$:  $\zeta(s)=\zeta(s,1)
= (2^s - 1)^{-1} \zeta(s,1/2)$.

The sum of Eq. (E.15) also has a close relation to $\psi^{(n)}(1/2)$, where 
$\psi^{(j)}$ is the polygamma function, because \cite{grad}
$$\psi^{(n)}\left ({1 \over 2}\right ) = (-1)^{n+1} n! \sum_{k=0}^\infty 
{1 \over {(k+1/2)^{n+1}}}.  \eqno(E.16a)$$
Then, with the use of Eq. (E.15), we have
$$\psi^{(n)}\left ({1 \over 2}\right )=(-1)^{n+1} n! (2^{n+1} - 1)\zeta(n+1),  
\eqno(E.16b)$$   
which is the expected result.  In general, we have $\psi^{(n)}(x)=(-1)^{n+1}n!
\zeta(n+1,x)$.

We may also write an integral representation for the polygamma function which
is very useful for evaluating terms in explicit formulas for sums over 
zeros of zeta functions.  By differentiating an integral representation for
$\psi(z)+\gamma$, we have
$$\psi^{(m-1)}(z)=(-1)^m\int_0^\infty {{e^{-zt}t^{m-1}} \over {1-e^{-t}}}dt
={{(-1)^m} \over 2}\int_0^\infty {{t^{m-1}e^{-(z-1/2)t}} \over {\sinh(t/2)}}
dt, \eqno(E.17)$$
giving the specific values
$$\psi^{(m-1)}\left({1 \over 2}\right)=(-1)^m 2^{m-1}\int_0^\infty 
{y^{m-1} \over {\sinh y}}dy, \eqno(E.18a)$$
$$\psi^{(m-1)}(1)=(-1)^m2^{m-1}\int_0^\infty {y^{m-1}e^{-y} \over {\sinh y}}
dy, \eqno(E.18b)$$
and
$$\psi^{(m-1)}\left({3 \over 2}\right)={{(-1)^m} \over 2}\int_0^\infty 
{{t^{m-1}e^{-t}} \over {\sinh(t/2)}} dt. \eqno(E.18c)$$

\pagebreak
\centerline{\bf Appendix EII:  Explicit Formulas for Dirichlet and Hecke 
L-Functions}
\medskip

Here we give alternative derivations of the very recent main results of Li 
\cite{li2}, Theorems 1 and 2, of Ref. \cite{li2}.  The procedure is very 
similar to the proof of Theorem 1 of the text.  The Riemann zeta function 
case extends since the Dirichlet and Hecke L-functions also have product 
expansions over their zeros and have explicit forms of their logarithmic 
derivatives.  These derivations also make it very apparent that certain 
polygamma constants are the source of the elementary sums described in the 
first part of this Appendix.

Due to the product expansion of $\xi(s,\chi)$, we have the formula  
$$\lambda_\chi(n)=\sum_{m=1}^n {n \choose m}{1 \over {(m-1)!}}\left [{{d^m} 
\over {ds^m}}\ln \xi(s,\chi) \right]_{s=1}.  \eqno(E.19)$$
This equation is the analog of Eq. (15) of the text or Eq. (G.5) of Appendix 
G for the Riemann zeta function case.  From Eq. (E.1) we have
$$\ln \xi(s,\chi)=-{{(s+a)} \over 2}\ln \left({\pi \over r}\right)+\ln \Gamma 
\left({{s+a} \over 2}\right )+\ln L(s,\chi), \eqno(E.20)$$
giving
$${d \over {ds}}\ln \xi(s,\chi)=\ln\left({r \over \pi}\right )+{1 \over 2}\psi
\left({{s+a} \over 2}\right )-\sum_{n=1}^\infty {{\Lambda(n)\chi(n)} \over n^s}, 
~~~~~~~~ \mbox{Re}~ s > 1, \eqno(E.21)$$
where $\psi=\Gamma'/\Gamma$ is the digamma function and $\Lambda$ is the von 
Mangoldt function, such that $\Lambda(k)=\ln p$ when $k$ is a power of a prime 
and $\Lambda(k)=0$ otherwise.  For $m \geq 2$, we then have
$$[\ln \xi(s,\chi)]^{(m)}={1 \over 2^m}\psi^{(m-1)}\left({{s+a} \over 2}\right )
-(-1)^{m-1} \sum_{n=1}^\infty {{\Lambda(n)\chi(n)} \over n^s} \ln^{m-1} n,  
\eqno(E.22)$$
where $\psi^{(n)}$ is again the polygamma function.  By taking the limit 
$s \to 1$ in Eq. (E.22) we then obtain the representation
$$\lambda_\chi(n)=\left[\ln\left({r \over \pi}\right)+\psi\left({{a+1} \over 2}
\right ) \right ]{n \over 2}+\sum_{m=2}^n {n \choose m} {1 \over {(m-1)!}}
2^{-m}\psi^{(m-1)} \left({{a+1} \over 2}\right )$$
$$-\sum_{m=1}^n{n \choose m}{{(-1)^{m-1}} \over {(m-1)!}}
\sum_{n=1}^\infty {{\Lambda(n)\chi(n)} \over n} \ln^{m-1} n,  \eqno(E.23)$$
where $\psi(1/2)= -\gamma-2\ln 2$, $\psi(1)=-\gamma$, $\gamma$ is the Euler 
constant, $\psi^{(m-1)}(1)=(-1)^m(m-1)!\zeta(m)$, and $\psi^{(m-1)}(1/2)$ is 
given in Eq. (E.16b).  The infinite series in the sum $S_\chi(n)$ is 
convergent by the prime number theorem for arithmetic progressions 
\cite{li2,davenport}.  We have therefore obtained the result Eq. (E.3).

Similarly, due to the product expansion of $\xi_E(s)$, we have the formula  
$$\lambda_E(n)=\sum_{m=1}^n {n \choose m}{1 \over {(m-1)!}}\left [{{d^m} 
\over {ds^m}}\ln \xi_E(s) \right]_{s=1},  \eqno(E.24)$$
where from Eq. (E.8) we have
$$\ln \xi_E(s)=\ln c_E + {s \over 2}\ln N-s\ln 2\pi+\ln\Gamma\left(s+{1 \over 
2} \right ) +\ln L_E\left(s+ {1 \over 2}\right),  \eqno(E.25)$$
and
$${d \over {ds}}\ln \xi_E(s)={1 \over 2}\ln N - \ln 2\pi + \psi
\left(s+{1 \over 2}\right)-\sum_{n=1}^\infty {{\Lambda(n)b(n)} \over n^{s+1/2}}, ~~~~~~~~
\mbox{Re}~ s > 1, \eqno(E.26)$$ %need to review and get a reference for HERE !!
where $b(n)$ is discussed in the first part of this Appendix.
For $m \geq 2$, we then have
$$[\ln \xi_E(s)]^{(m)}=\psi^{(m-1)}\left(s+{1 \over 2}\right )
-(-1)^{m-1} \sum_{n=1}^\infty {{\Lambda(n)b(n)} \over n^{s+1/2}} \ln^{m-1} n. 
\eqno(E.27)$$
By taking the limit $s \to 1$ in Eq. (E.27) we then obtain the representation
$$\lambda_E(n)=\left[\ln\left({\sqrt{N} \over {2\pi}}\right)+\psi\left({3 
\over 2} \right) \right]n + \sum_{m=2}^n {n \choose m} {1 \over {(m-1)!}}
\psi^{(m-1)} \left({3 \over 2}\right)$$
$$-\sum_{m=1}^n {n \choose m}{{(-1)^{m-1}} \over {(m-1)!}}
\sum_{n=1}^\infty {{\Lambda(n)b(n)} \over n^{3/2}} \ln^{m-1} n,  \eqno(E.28)$$
where $\psi(3/2)= 2(1-\ln 2)-\gamma$ and $\psi^{(m-1)}(3/2)=(-1)^m(m-1)!
[2^m(\zeta(m)-1)-\zeta(m)]$, giving the result Eq. (E.12).
%Could use a remark here too on convergence of the infinite sum in Eq. (E.28)

Finally, we consider the case of the Dedekind zeta function $\zeta_k$, for 
which we need to introduce some additional notation.  We let $k$ be an 
algebraic number field with $r_1$ real places, $r_2$ imaginary places, and 
degree $\tilde{n} = r_1+2r_2$.  The zeta function $\zeta_k$ has the product 
expansion $\zeta_k(s)=\prod_p (1-Np^{-s})^{-1}$ for Re $s>1$, where the 
product is taken over all finite prime divisors of $k$.  We put 
$G_1(s) = \pi^{-s/2}\Gamma(s/2)$ and $G_2(s) = (2\pi)^{1-s} \Gamma(s)$, 
so that obviously $G_1(1) = G_2(1) = 1$.  Then the function
$$Z_k(s) \equiv G_1^{r_1}(s)G_2^{r_2}(s)\zeta_k(s), \eqno(E.29)$$
satisfies the functional equation $Z_k(s)=|d_k|^{1/2-s}Z_k(1-s)$, where $d_k$ 
is the discriminant of $k$.  

We let $c_k=2^{r_1}(2\pi)^{r_2}hR/e$, where $h$, $R$, and $e$ are respectively
the number of ideal classes of $k$, the regulator of $k$, and the number of 
roots of unity in $k$.  With
$$\xi_k(s) \equiv c_k^{-1}s(s-1)|d_k|^{s/2}Z_k(s), \eqno(E.30)$$
this function is entire and has $\xi_k(0)=1$ \cite{weil,neukirch}.  
We first present a motivation that an explicit formula analogous to that for 
$\lambda_E$ and $\lambda_\chi$ exists, and then develop the corresponding 
explicit formula, putting
$$\lambda_n=\sum_{m=1}^n {n \choose m}{1 \over {(m-1)!}}\left [{{d^m} \over 
{ds^m}}\ln \xi_k(s) \right]_{s=1}.  \eqno(E.31)$$

From Eqs. (E.29) and (E.30) we have
$$\ln \xi_k(s)=-\ln c_k + \ln s + \ln (s-1)+{s \over 2}\ln |d_k|+r_1\left[-{s 
\over 2}\ln \pi -\ln\Gamma\left({s \over 2}\right )\right ]$$
$$+r_2[(1-s)\ln(2\pi) + \ln \Gamma (s)]+\ln \zeta_k(s),  \eqno(E.32)$$
and
$${d \over {ds}}\ln \xi_k(s)={1 \over s}+{1 \over {s-1}} + {1 \over 2}
\ln |d_k|+{r_1 \over 2}\left[-\ln \pi + \psi\left({s \over 2}\right )
\right ]+r_2[-\ln(2\pi)+\psi(s)]+{{\zeta_k'(s)} \over {\zeta_k(s)}}, 
~~~~\mbox{Re}~ s > 1, \eqno(E.33)$$ 
where \cite{amo1989}
$${{\zeta_k'(s)} \over {\zeta_k(s)}} = -\sum_p\sum_{m=1}^\infty {{\ln Np} 
\over {Np^{ms}}}, ~~~~\mbox{Re}~ s > 1. \eqno(E.34)$$ 
In Eq. (E.34), $p$ runs over the prime ideals of $k$ and $N$ represents the
norm.

For $m \geq 2$, we then have
$${d^m \over {ds^m}}\ln \xi_k(s)={{(-1)^m(m-1)!} \over s^{m-1}}+{{(-1)^m(m-1)!} 
\over {(s-1)^{m-1}}} + {r_1 \over 2^m}\psi^{(m-1)}\left({s \over 2}\right )
+ r_2 \psi^{(m-1)}(s)$$
$$+{d^{m-1} \over {ds^{m-1}}}\left[{{\zeta_k'(s)} \over 
{\zeta_k(s)}} \right ],~~~~\mbox{Re}~ s > 1, \eqno(E.35)$$ 
where the evaluation of the first, third and fourth terms on the right side 
of Eq. (E.35) at $s=1$ gives the contribution to $\lambda_n$ of 
$$\lambda_n^{(\psi)}=\sum_{m=2}^n (-1)^m{n \choose m} \left\{1+[(1-2^{-m})r_1
+r_2] \zeta(m)\right \}.  \eqno(E.36)$$
The evaluation of all of the terms on the right side of Eq. (E.33) but
the second and last at $s=1$ gives to $\lambda_n$ the contribution
$n[1+{1 \over 2}\ln|d_k|-{\tilde{n} \over 2}(\ln \pi+\gamma)-(r_1+r_2)\ln 2]$.
With the aid of Eq. (E.34) we have
$${d^{m-1} \over {ds^{m-1}}}\left[{{\zeta_k'(s)} \over 
{\zeta_k(s)}} \right ] = -\sum_p\sum_{\ell=1}^\infty (-1)^{m-1} \ell^{m-1}
{{\ln^m Np} \over {Np}^{\ell s}}, ~~~~\mbox{Re}~ s > 1. \eqno(E.37)$$ 
Taking the limit $s \to 1$ in Eqs. (E.33)--(E.35) and (E.37) should yield
the final explicit representation for $\lambda_n$, subject to
justification of the convergence of the resulting series.

%next will be explicit formula development ... TBS ...


appendixf.tex
\centerline{\bf Appendix F:  Further Riemann zeta function sum estimations}
\medskip

As a generalization of sums such as $S_1$ of Eq. (16) of the text and $S_0$
of Eq. (A.15) of Appendix A, we consider here sums of the form
$$S_\nu(\kappa,n) \equiv \sum_{m=\nu+2}^n (-1)^m {n \choose m}\kappa^m
\zeta(m-\nu), \eqno(F.1)$$
where $\nu +2 \leq n$, which can be extended to $|\mbox{Re} ~\nu|+2 \leq n$.
In Eq. (F.1) we have introduced both the positive multiplier $\kappa$ and shift 
$\nu$.  The special cases of Eq. (F.1) of direct interest to this paper are
$\nu=0$ with $\kappa=1$ or $\kappa = 2^{\pm 1}$.  %what's the range for \kappa?
We are interested to both reformulate the sum $S_\nu$ and to obtain a
lower bound for it.

If we reorder the two sums in Eq. (F.1), the inner sum takes the form
$$\sum_{m=\nu+2}^n(-1)^m{n \choose m}{\kappa^m \over j^{m-\nu}}
=(-1)^\nu {{\kappa^{\nu+2} n!} \over j^2}\sum_{m=0}^{n-\nu-2} {{(1)_m} \over
{(n-m-\nu-2)!(m+\nu+2)!}}{1 \over {m!}}\left({\kappa \over j} \right )^m .
\eqno(F.2)$$
%check for need for a say (-1)^?? factor just below in (.)_j relations:  ...
If we use the relations $(m+\nu+2)!=(\nu+2)!(\nu+3)_m$ and
$(n-m-\nu-2)! = (n-\nu-2)!/(2+\nu-n)_m$, where the Pochhammer symbol
$(z)_n=\Gamma(z+n)/\Gamma(z)$, we obtain the terminating hypergeometric form
$$\sum_{m=\nu+2}^n(-1)^m{n \choose m}{\kappa^m \over j^{m-\nu}}
= (-1)^\nu {n \choose {\nu+2}} {\kappa^{\nu+2} \over j^2}
~_2F_1(1,2-n+\nu;\nu+3;\kappa/j),  \eqno(F.3)$$
giving
$$S_\nu(\kappa,n)=
(-1)^\nu {n \choose {\nu+2}} \kappa^{\nu+2} \sum_{j=1}^\infty {1 \over j^2}
~_2F_1(1,2-n+\nu;\nu+3;\kappa/j).  \eqno(F.4)$$

We now define the comparison integral
$$I_\nu(\kappa,n) \equiv
(-1)^\nu {n \choose {\nu+2}} \kappa^{\nu+2} \int_1^\infty {1 \over j^2}
~_2F_1(1,2-n+\nu;\nu+3;\kappa/j) dj.  \eqno(F.5)$$
With the change of variable $v=1/j$ the integration is easily accomplished
\cite{grad} in terms of the generalized hypergeometric function $_pF_q$
\cite{pfqref}:
$$I_\nu(\kappa,n) 
= (-1)^\nu {n \choose {\nu+2}} \kappa^{\nu+2} ~_3F_2(1,1,\nu+2-n;2,\nu+3;
\kappa).  \eqno(F.6)$$
%need to give the _3F_2 reduction formula and reference ...................
Equation (F.6) is simply the result of term-by-term integration and the
fact that $(1)_k/(2)_k = 1/(k+1)$.
The relation $S_\nu(\kappa,n) \geq I_\nu(\kappa,n)$ then yields a family
of inequalities.

When $\kappa$ is unity we have the reduction $\psi(z)=(z-1) ~_3F_2(1,1,2-z;
2,2;1)-\gamma$ and therefore 
$$I_\nu(1,n) = (-1)^\nu {n \choose {\nu+1}} [\psi(n+1)+\gamma - H_{\nu+1}],
\eqno(F.7)$$
where $\psi$ is the digamma function and $H_n=\sum_{k=1}^n 1/k$ is the $n$th 
harmonic number \cite{cofjcam1}, which can also be written as
$$I_\nu(1,n) = (-1)^\nu {n \choose {\nu+1}} [H_n - H_{\nu+1}].  \eqno(F.8)$$ 
For $\nu=0$, this gives the result of Eq. (A.17).
An alternative form of $I_\nu(1,n)$ can be obtained by applying Theorem 1 of
Ref. \cite{rao}:
$$I_\nu(1,n)=(-1)^\nu {n \choose {\nu+1}} \left [\psi(n-\nu)-\psi(\nu+2)
-\sum_{k=1}^{\nu +1} {{(-\nu-1)_k} \over {k(n-\nu)_k}} \right ], \eqno(F.9)$$
valid for $n>0$ and integral $\nu \geq -1$.  The relations above at unit
argument can also be looked upon as special cases of \cite{luke}
$$_3F_2(1,1,\nu+1;2,\lambda+1;1)={\lambda \over \nu}[\psi(\lambda)-\psi(
\lambda-\nu)], ~~~~\nu \neq 0, \mbox{Re}(\lambda-\nu)>0, $$
$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\lambda\psi'(\lambda),
~~~~\nu=0, ~~~~\mbox{Re}(\lambda)>0,  \eqno(F.10)$$
i.e.,
$$\sum_{n=1}^\infty {{(\nu)_n} \over {n(\lambda)_n}}=\psi(\lambda)-\psi(
\lambda-\nu), ~~~~\mbox{Re}(\lambda-\nu)>0, ~~~~\lambda\neq 0, -1, -2, \ldots.
\eqno(F.11)$$

For $\kappa=1/2$ we obtain %to verify ......
$$I_\nu(1/2,n) 
= (-1)^\nu {n \choose {\nu+2}} {1 \over 2^{\nu+2}} ~_3F_2(1,1,\nu+2-n;2,\nu+3;
1/2),  \eqno(F.12)$$
which for $\nu=0$ is in agreement with $I_0-I_1$, where $I_1$ is given in 
Eq. (24) of the text.
For $\kappa=2$ we obtain %to verify ......
$$I_\nu(2,n) 
= (-1)^\nu {n \choose {\nu+2}} 2^{\nu+2} ~_3F_2(1,1,\nu+2-n;2,\nu+3;
2),  \eqno(F.13)$$ 
%put correct equ. (A.22) in here, e.g. with extra \nu-dependent factors ...
which is in agreement with $I_3$ of Eq. (A.22) of Appendix A when $\nu=0$.
%would like to get the more general explicit form with \kappa \psi(n) + ...
%a to do exercise ...

appendixg.tex
\centerline{\bf Appendix G:  Alternative representation of Li's $\lambda_j$'s}
\medskip

Here we extend Theorem 1, based upon an expansion of the logarithmic 
derivative of the Riemann zeta function with a larger radius of convergence
than Eq. (12) of the text.
We demonstrate the following representation,
$$\lambda_n=-\sum_{m=1}^n {n \choose m} \eta^{(12)}_{m-1}
+7-\left [\left({2 \over 3}\right )^n+\left({4 \over 5}\right)^n+\left (
{6 \over 7}\right )^n+\left({8 \over 9}\right )^n+\left({{10} \over {11}}
\right)^n+\left({{12} \over {13}}\right)^n\right ]$$
$$+\sum_{m=2}^n (-1)^m {n \choose m} (1-2^{-m})\zeta(m)-{n \over 2} (\gamma
+\ln \pi + 2\ln 2), \eqno(G.1)$$
where we currently do not have an arithmetic interpretation of the constants 
$\eta^{(12)}_j$.   

From the expansion around $s=1$ of the logarithmic derivative of the
zeta function, 
$${{\zeta'(s)} \over {\zeta(s)}} = -{1 \over {(s-1)}} + \sum_{j=1}^6 {1 \over 
{(s+2j)}} - \sum_{p=0}^\infty \eta^{(12)}_p (s-1)^p, \eqno(G.2)$$ 
we have
$$\ln \zeta(s)=-\ln(s-1)+\sum_{j=1}^6 \ln (s+2j)
-\sum_{p=1}^\infty {{\eta^{(12)}_{p-1}} \over p}(s-1)^p
+ \mbox{constant},  \eqno(G.3)$$
giving
$$\ln \xi(s)=-\ln 2 +\ln s-{s \over 2}\ln \pi+\ln \Gamma\left({s \over 2}
\right) + \mbox{constant} + \sum_{j=1}^6 \ln (s+2j)
-\sum_{p=1}^\infty {\eta^{(12)}_{p-1} \over p}(s-1)^p.  \eqno(G.4)$$
With the expansion (G.2) the radius of convergence has been increased to 13
(see Figure 1),
%review ... include in Figure 1 the approximate location of \rho_1, the
%first complex zero
as we have included the contribution of all trivial zeros of $\zeta$ prior
to the encounter with the first complex zero $\rho_1$. 
We next evaluate
$$\lambda_n=\sum_{m=1}^n {n \choose m}{1 \over {(m-1)!}}\left [{{d^m} \over 
{ds^m}}\ln \xi(s) \right]_{s=1},  \eqno(G.5)$$
using again the special values $\psi(1/2)=-\gamma-2 \ln 2$ and
$\psi^{(n)}(1/2)=(-1)^{n+1}n!(2^{n+1}-1)\zeta(n+1)$ for $n \geq 1$, where
$\psi=\Gamma'/\Gamma$ is the digamma function and $\psi^{(j)}$ is the
polygamma function.  Recalling the relations
$${{d^j} \over {ds^j}} \ln (s+2k)=-{{(-1)^j (j-1)!} \over {(s+2k)^j}},
~~~~~~~~ j \geq 1, \eqno(G.6)$$
and $(d^j/ds^j) (s-1)^k = 0$ for $k < j$, and the sum
$$\sum_{m=1}^n {n \choose m}\left (- {1 \over k} \right )^m 
=-1 + \left ({{k-1} \over k} \right )^n, \eqno(G.7)$$
we find Eq. (G.1).
%So what other conclusions for a lower bound to \lambda_n can I now draw
%rigorously?   to be considered more ...............................
The constant term of $6=7-1$ in Eq. (G.1) serves as a count of the number 
of trivial zeros of $\zeta$ accounted for in the expansion (G.2) while the 
additional explicit negative terms beyond Eq. (10) of the text appearing 
there are exponentially decreasing with $n$.
We have developed in Eq. (G.2) an expansion with coefficients
$\eta_j^{(12)}$ whose magnitudes increase no faster than $1/13^j$ for
large $j$.

By using Eq. (20) for $S_1$, an extension of the Corollary of the text is
\newline{\bf Corollary G1}
$$\lambda_n \geq {n \over 2} \ln n -(1+\ln \pi + 2\ln 2){n \over 2} + {{15}
\over 2}$$
$$-\left [\left({2 \over 3}\right )^n+\left({4 \over 5}\right)^n
+\left ({6 \over 7}\right )^n+\left({8 \over 9}\right )^n+\left({{10} \over 
{11}}\right)^n+\left({{12} \over {13}}\right)^n\right ]-|S_2^{(12)}|, 
\eqno(G.8)$$
where we have put $S_2^{(12)} \equiv -\sum_{m=1}^n {n \choose m}
\eta_{m-1}^{(12)}$.

In developing expansions such as
$$\zeta(s->-2k)=\zeta'(-2k)(s+2k)+{1 \over 2}\zeta''(-2k)(s+2k)^2+{1 \over 6}
\zeta'''(-2k)(s+2k)^3 + O[(s+2k)^4], ~~~~~~~~k \geq 1, \eqno(G.9)$$
and
$${{\zeta'(s)} \over {\zeta(s)}} = {1 \over {(s+2k)}}+{{\zeta''(-2k)} \over
{2\zeta'(-2k)}}+\left[-{{[\zeta''(-2k)]^2} \over {4[\zeta'(-2k)]^2}} +
{{\zeta'''(-2k)} \over {3\zeta'(-2k)}} \right ] (s+2k) + O[(s+2k)^2],
\eqno(G.10)$$
it is useful to have the derivatives
$$\zeta'(-2n)=(-1)^n{{(2n)!} \over {2(2\pi)^{2n}}} \zeta(2n+1), 
~~~~~~~~ n \geq 1, \eqno(G.11)$$
and
$$\zeta''(-2n)=(-1)^n{{(2n)!} \over {2(2\pi)^{2n}}} \left\{\left[\ln(4\pi^2) 
-2\psi(2n)-{1 \over n}\right ] \zeta(2n+1) - 2\zeta'(2n+1)\right \},
~~~~~~n \geq 1, \eqno(G.12)$$
which follow easily by differentiating the functional equation for $\zeta(s)$
and putting $s=2n+1$.

%Figure to go in here (TBS) ...................
%with the approximate location of the first nontrivial zero $\rho_1$ indicated.
%I should maybe include more tabulated etas and lambdas in Appendix D,
%at least to n=50 and perhaps to 75 or 100 now .................
%I need to get the Apostol and xxx Math. Comp. (1985) and (19xx) papers
%and understand .... in particular the zeta derivative results there ....
%also continue to read the Voros (2002) paper ...............
%and where have I put the 1993 Varga et al. paper on the use of Lehmer zeros?

%In reference to Appendix E, could any of this be generalized to extend the
%representations there for the other lambdas, lambda_E, lambda_\chi, etc.??
We note in passing the numerical value of the constant
$${1 \over 2}\sum_{k=1}^6 {{\zeta''(-2k)} \over {\zeta'(-2k)}}=-{{81959} \over 
{5544}} + 6(\gamma+\ln 2\pi)-\sum_{k=1}^6 {{\zeta'(2k+1)} \over {\zeta(2k+1)}}
\simeq -0.0926073.  \eqno(G.13)$$
We see from Eq. (G.2) that the value of $\eta_0^{(12)}$ is given by
$$\eta_0^{(12)} = \gamma - \sum_{k=1}^6 {1 \over {2k+1}} \simeq 0.377918.
\eqno(G.14)$$
Therefore we can modify Eq. (G.8) to
{\newline \bf Corollary G2}
$$\lambda_n \geq {n \over 2} \ln n -(1+\ln \pi + 2\ln 2){n \over 2} + {{15}
\over 2}$$
$$-\left [\left({2 \over 3}\right )^n+\left({4 \over 5}\right)^n
+\left ({6 \over 7}\right )^n+\left({8 \over 9}\right )^n+\left({{10} \over 
{11}}\right)^n+\left({{12} \over {13}}\right)^n\right ]-\left|\eta_0^{(12)}
\right|n-\left|\sum_{m=2}^n {n \choose m} \eta_{m-1}^{(12)}\right|. 
\eqno(G.15)$$



appendixh.tex
\centerline{\bf Appendix H:  On derivatives of the Riemann zeta function}
\medskip

Here we capture various formulas for integer order derivatives of the
zeta function.  We anticipate that these could be useful in further
development of the discrete moment problem for the coefficients 
$\eta_j$ of Eqs. (10)-(14) and Eq. (41) and $\eta_j^{(12)}$ of Eqs. 
(G.1)-(G.4) of Appendix G.

We first note that the functional equation for $\zeta$, along with the 
evaluation $\zeta(1-2n)=-B_{2n}/2n$ for $n \geq 1$, where $B_n$ are 
Bernoulli numbers, yields
$$\zeta'(-1)={{\zeta'(2)} \over {2\pi^2}}+{1 \over {12}}(\ln 2\pi + \gamma),
\eqno(H.1)$$
and 
$$\zeta''(-1)={1 \over 6}\left({\pi^2 \over 8}-{{\ln^2 2} \over 2}\right ) 
+\left[(1-\gamma){\pi^2 \over 6}+\zeta'(2)\right ]{{\ln 2} \over \pi^2}
-{{(1-\gamma)} \over \pi^2}\zeta'(2) -{1 \over {2\pi^2}}\zeta''(2)$$
$$-{1 \over {12}}\left[-1+(1-\gamma)^2+{\pi^2 \over 6}\right ]+{1 \over {12}}
\ln^2 \pi + 2\ln \pi \zeta'(-1), \eqno(H.2)$$
where $\gamma$ is the Euler constant, and this can be continued to higher 
order derivatives.  The derived functional equation upon which Eqs. (H.1) and 
(H.2) are based is
$$2^{1-s}\Gamma(s)\zeta(s)\cos\left({\pi \over 2}s\right)[\psi(s)-\ln 2]
+2^{1-s}\Gamma(s)\zeta'(s)\cos\left({\pi \over 2}s\right)$$
$$-{\pi \over 2}2^{1-s}\Gamma(s)\zeta(s)\sin\left({\pi \over 2}s\right)
=\pi^s \ln \pi \zeta(1-s) - \pi^s \zeta'(1-s).  \eqno(H.3)$$
Now Elizalde \cite{elizalde} has given an expression for $\zeta'(-m,q)$, 
where $\zeta(z,q)$ is the Hurwitz zeta function, valid for any negative 
integer value of $z$.  We state here special cases for $q=1$:
$$\zeta'(-1)=-{1 \over 6}-{1 \over 2}\sum_{k=1}^\infty {{B_{2k+2}} \over
{k(2k+1)(2k+2)}}, \eqno(H.4)$$
$$\zeta'(-2)=-{1 \over {36}}-\sum_{k=1}^\infty {{B_{2k+2}} \over
{(2k-1)k(2k+1)(2k+2)}}, \eqno(H.5)$$
and
$$\zeta'(-k)=-{1 \over {(k+1)^2}}-\sum_{\ell=1}^\infty {{(-1)^\ell (2\ell
-1)!} \over {2^{2\ell-1}\pi^{2\ell}}}\left [\sum_{r=0}^{min(2\ell-2,k)}
{k \choose r}{{(-1)^r} \over {(2\ell-r-1)}}\right ]\zeta(2\ell).  \eqno(H.6)$$
%recall p. 1590 of Yue and K. S. Wms, 1993.  the min(.,.) serves to combine
%the two forms of a_{2k} in Elizalde's Eq. (16).  the B_{2k+2} are not
%explicit by using \zeta(2k), ...
This means that we also have explicit expressions for $\zeta'(1+k)$.  In
particular Eqs. (H.1) and (H.4) or (H.6) yield an explicit form for 
$\zeta'(2)$.

On the other hand, we can employ the integral representation, Eq. (38) of 
the text, to at least partially yield explicit values of the zeta
derivatives for Re $s > 1$.  Other integral representation could be used for
Re $s > 0$, but Eq. (38) serves for illustration.  In the following,
$\psi$ denotes the digamma function and $\psi^{(j)}$ the polygamma function,
as usual.

We have
$$\zeta'(s)=-\psi(s)\zeta(s)+{1 \over {\Gamma(s)}}\int_0^\infty {{t^{s-1}
\ln t} \over {e^t - 1}} dt, \eqno(H.7)$$
$$\zeta''(s)=-\psi'(s)\zeta(s)+\psi^2(s)\zeta(s)-2{{\psi(s)} \over {\Gamma(s)}}
\int_0^\infty {{t^{s-1}\ln t} \over {e^t-1}}dt+{1 \over {\Gamma(s)}}
\int_0^\infty {{t^{s-1} \ln^2 t} \over {e^t-1}}dt, \eqno(H.8)$$
and therefore
$${{\zeta'(s)} \over {\zeta(s)}}=-\psi(s)+{1 \over {\Gamma(s)\zeta(s)}}
\int_0^\infty {{t^{s-1} \ln t} \over {e^t - 1}} dt, \eqno(H.9)$$
and
$$\left[{{\zeta'(s)} \over {\zeta(s)}}\right]'=-\psi'(s)+{1 \over {\Gamma(s)
\zeta(s)}} \left\{\int_0^\infty {{t^{s-1} \ln^2 t} \over {e^t - 1}} dt
-{1 \over {\Gamma(s)\zeta(s)}} \left [\int_0^\infty {{t^{s-1} \ln t} \over 
{e^t - 1}} dt \right ]^2 \right \}.  \eqno(H.10)$$
This process can be continued,
$$\left[{{\zeta'(s)} \over {\zeta(s)}}\right]''=-\psi''(s)+\psi^3(s)-\psi(s)
\psi'(s)-\psi'(s){{\zeta'(s)} \over {\zeta(s)}}+\psi^2(s){{\zeta'(s)} \over
{\zeta(s)}}$$
$$+{1 \over {\Gamma(s)\zeta(s)}}[\psi'(s)-\psi^2(s)]
\int_0^\infty {{t^{s-1} \ln t} \over {e^t - 1}} dt
-{3 \over {\Gamma^2(s)\zeta^2(s)}} \int_0^\infty {{t^{s-1} \ln t} \over 
{e^t - 1}} dt \int_0^\infty {{t^{s-1} \ln^2 t} \over {e^t - 1}} dt$$
$$+{1 \over {\Gamma(s)\zeta(s)}}\int_0^\infty {{t^{s-1} \ln^3 t} \over 
{e^t - 1}} dt +{2 \over {\Gamma^3(s)\zeta^3(s)}}\left[\int_0^\infty {{t^{s-1} 
\ln t} \over {e^t - 1}} dt \right ]^3 , \eqno(H.11)$$
and we note that
$$-\psi^{(j)}(2)=(-1)^j j![\zeta(j+1)-1].  \eqno(H.12)$$

Following on Eq. (H.7) we have
$$\zeta^{(j+1)}(s)=-\sum_{m=0}^j{j \choose m}\psi^{(j-m)}(s)\zeta^{(m)}(s)
+\sum_{m=0}^j {j \choose m}\left[\left({d \over {ds}}\right)^m {1 \over 
{\Gamma(s)}} \right ]\int_0^\infty {{t^{s-1}\ln^{j-m+1} t} \over {e^t-1}} dt.  
\eqno(H.13)$$
%later would like to give the general formula (with the appropriate
%summation) for say the jth derivative of the logarithmic derivative of an
%arbitrary differentiable function ...
%would be desirable to include the general derivative d^m/ds^m 1/\Gamma
%in Eq. (H.12) too.

The relations of this Appendix can be developed much more for application to 
Eq. (41) of the text or elsewhere.



appendixi.tex
\centerline{\bf Appendix I:  A digamma function integral and a Mellin transform}
\medskip

Here we consider the integral
$$I(\lambda)=\int_1^\infty {{(1-x^{-\lambda})}\over {x(x^2-1)}} dx
\eqno(I.1)$$
of Eq. (3.4) of Ref. \cite{bombieri} and evaluate it in two different ways
from tabulated results \cite{grad}.
In Ref. \cite{bombieri} the asymptotic behaviour of this integral for
large $\lambda$ was of interest for determining a certain limit denoted by
the PF operation [Eq. (3.2) there].  We also note a polynomial of
Ref. \cite{bombieri} that can be written as a terminating confluent
hypergeometric series.  This associated Laguerre polynomial was used in 
calculating forward and inverse Mellin transforms and we give alternative 
transforms.

With the change of variable $v(x)=x^{-1}$ in Eq. (I.1) we have
$$I(\lambda)=\int_0^\infty {{v(1-v^\lambda)} \over {(1-v^2)}} dv.  
\eqno(I.2)$$
From Ref. \cite{grad} we then obtain
$$I(\lambda)={1 \over 2}\left[\psi \left({\lambda \over 2}+1\right) + \gamma
\right ]
={1 \over 2}\left[\psi \left({\lambda \over 2}\right) + \gamma + {2 \over
\lambda} \right ], ~~~~~~\lambda > -2,  \eqno(I.3)$$ 
where $\psi=\Gamma'/\Gamma$ is the digamma function.
On the other hand, we may employ a partial fractional decomposition in
Eq. (I.2) and another tabulated result \cite{grad} so that
$$I(\lambda)={1 \over 2}\int_0^1(1-v^\lambda)\left[{1 \over {(1-v)}}
-{1 \over {(1+v)}}\right ]dv
={1 \over 2}\left[\psi(\lambda +1)+\gamma-\ln 2+\int_0^1 {v^\lambda \over
{1+v}} dv \right ], \eqno(I.4)$$
where the last integral is given by \cite{grad}
$$\beta(\lambda+1)={1 \over 2}\left[\psi\left({\lambda \over 2}+1\right)
-\psi\left({{\lambda+1} \over 2}\right )\right ].  \eqno(I.5)$$
Use of the doubling formula $\psi[(\lambda+1)/2]=2\psi(\lambda)-\psi
(\lambda/2)-2\ln 2$ then again yields Eq. (I.3).  In Appendix A we have 
additionally given many inequalities for the digamma function.

The polynomial
$$P_n(x)=\sum_{j=1}^n {n \choose j} {x^{j-1} \over {(j-1)!}} \eqno(I.6)$$
was used in Ref. \cite{bombieri} in connection with computing Mellin 
transforms.  By using the relations
$${n \choose j}={{(-1)^j(-n)_j} \over {j!}} ~~~~ \mbox{and} ~~~~
(-n)_{j+1}=-n(1-n)_j, \eqno(I.7)$$
where $(.)_n$ is the Pochhammer symbol, this polynomial can be written as a 
terminating confluent hypergeometric series:  $P_n(x)=n ~_1F_1(1-n;2;-x)$.
In particular, a certain Mellin transform involving $P_n$ converts to a
Laplace transform:
$$\int_0^1P_n(\ln x)x^{s-1}dx=\int_0^\infty P_n(-u)e^{-su} du
=n\int_0^\infty ~_1F_1(1-n;2;u)e^{-su} du$$  
$$={n \over s} F\left(1-n,1;2;{1 \over s}\right)=1-\left(1-{1 \over s}
\right )^n, \eqno(I.8)$$
where $F$ is the Gauss hypergeometric function \cite{grad},
which is the expected result \cite{bombieri}.  In obtaining Eq. (I.8) we
have used the reduction \cite{grad}
$$F\left(1-n,1;2;-{z \over t}\right)={{(t+z)^n-t^n} \over {nzt^{n-1}}}.
\eqno(I.9)$$

Another useful point of view of the particular polynomial (I.6) is afforded
by the theory of Laguerre polynomials $L_n$.  This family is orthogonal on the
interval $[0,\infty)$ with decaying exponential weight function.  We have the
relations
$$P_n(-x)=-{{dL_n(x)} \over {dx}} = L_{n-1}^1 (x), \eqno(I.10)$$
where $L_n^\alpha$ is an associated Laguerre polynomial.  In addition, the
recursion relations satisfied by the Laguerre polynomials \cite{grad} give
$$P_n(-x)=-{n \over x}[L_n(x)-L_{n-1}(x)]=-{{(n+1)} \over x}L_{n+1}(x)+
{{(n+1-x)} \over x}L_n(x).  \eqno(I.11)$$
Then one can recast the important Mellin transform-inverse transform pair of
Eq. (I.8):
$$\int_0^\infty P_n(-u)e^{-su} du=-\int_0^\infty {{dL_n(u)} \over {du}}e^{-su}
du=1-\left(1-{1 \over s}\right )^n, ~~~~~~\mbox{Re} ~s > 0. \eqno(I.12)$$
In obtaining this equation, one can use integration by parts, the Laplace 
transform of $L_n$ \cite{grad}, and the property $L_n(0)=1$.
All these relations are consistent with the connection $L_{n-1}^1(x)=n ~_1F_1
(1-n;2;x)$ and the derivative property of the confluent hypergeometric function.
appendixj.tex
\centerline{\bf Appendix J: Regarding Conjectures 1--3}
\medskip

We present here plausibility arguments in possible support of our
conjectures concerning the detailed behaviour of the sequences $\{\sigma_k\}$
and $\{\lambda_j\}$.  
For this discussion, we let $N(T)$ be the number of zeros of the Riemann
zeta function in the critical strip in the upper half plane to height $T$.  
That is, $N(T)$ denotes the number of complex zeros in the rectangle $0 
\leq \mbox{Re}~s \leq 1$ and $0 \leq \mbox{Im}~s \leq T$.

Backlund \cite{backlund} showed that $N(T)$ satisfies
$$N(T)={T \over {2\pi}}\ln\left({T \over {2\pi}}\right)-{T \over {2\pi}}
+{7 \over 8}+e(T), \eqno(J.1)$$
where
$$|e(T)|<0.137 \ln T+0.443 \ln \ln T + 4.35 ~~~~~~ \mbox{for}~~ T \geq 2.
\eqno(J.2)$$
We believe then that if one were to assume certain statistical properties
of the distribution of the Riemann zeros, Conjectures 2 and 3 would follow.

From Eqs. (J.1) and (J.2) we can show that there is a constant $T_0$ 
such that
$$\pi [N(T+1)-N(T)] \leq \pi \ln T  ~~~~~~\mbox{for}~~ T \geq T_0.  
\eqno(J.3)$$
If we write $N(T)=M(T)+e(T)$, then
$$M(T+1)-M(T) ={1 \over {2\pi}}\ln\left({T \over {2\pi}}\right)+{1 \over
{2\pi}}\left({1 \over {2T}}-{1 \over {6T^2}}+{1 \over {12T^3}} - \cdots
\right)<{1 \over {2\pi}}\ln\left({T \over {2\pi}}\right) +{1 \over {4\pi T}}.
\eqno(J.4)$$
Then with Eq. (J.2) we have
$$N(T+1)-N(T)<{1 \over {2\pi}}\ln\left({T \over {2\pi}}\right)+{1 \over {
4\pi T}}0.137\left[2\ln T+\ln\left(1+{1 \over T}\right)\right]+0.866 \ln
\ln(T+1)+8.70,$$ 
$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~T \geq 2.  \eqno(J.5)$$
Since $\ln(1+1/T)<1/T$, we find
$$N(T+1)-N(T)<0.433 \ln T+0.866\ln\ln(T+1)+8.407+{{0.216} \over T}.  
\eqno(J.6)$$
We also have that
$$0.866\ln\ln (T+1) \leq 0.135 \ln T~~~~\mbox{for}~~T \geq T_0, \eqno(J.7)$$
$$8.407 \leq 0.431 \ln T ~~~~\mbox{for}~~T \geq T_0, \eqno(J.8)$$
and we may take $T_0=3 \times 10^8$.
These relations yield inequality (J.3).

We now assume that the Riemann hypothesis holds and consider one possible
bound that may result for the sums $\sigma_k=\sum_j \rho_j^{-k}$, where
$\{\rho_j\}$ represents the nontrivial zeros of the zeta function.
The nontrivial zeros have the form $\rho_j=1/2+\epsilon+i\alpha_j$,
where bounds for $\epsilon$ exist in the literature due to results on
zero-free regions.  As mentioned in the text, a zero $\rho_j$ enters the
sum $\sigma_k$ along with its complex conjugate.  We then consider the sums
$$\sum_{j=m}^\infty {1 \over {\alpha_j^k}} \leq {1 \over {(k-1)^2}}
{{[1-\ln([\alpha_m]-1)+k\ln([\alpha_m]-1)]} \over {([\alpha_m]-1)^{k-1}}},
~~~~~~ [\alpha_m] \geq T_0, \eqno(J.9)$$
as an approximation to $\sigma_k$, where $[x]$ denotes the greatest integer 
contained within $x$.  We have
$$\sum_{j=m}^\infty {1 \over {\alpha_j^k}} \leq \sum_{j=[\alpha_m]}^\infty
\sum_{j \leq \alpha_\ell<j+1}{1 \over \alpha_\ell^k}$$
$$\leq \sum_{j=[\alpha_m]}^\infty {{[N(j+1)-N(j)]} \over j^k}\leq 
\sum_{j=[\alpha_m]}^\infty {{\ln j} \over j^k}, ~~~~\mbox{for}~~[\alpha_m]
\geq T_0, \eqno(J.10)$$
where we applied inequality (J.3).  Since the last sum in inequality (J.10)
is bounded by
$$\int_{[\alpha_m]-1}^\infty {{\ln u} \over u^k}du = {1 \over {(k-1)^2}}
{{[1-\ln([\alpha_m]-1)+k\ln([\alpha_m]-1)]} \over {([\alpha_m]-1)^{k-1}}},
\eqno(J.11)$$
we obtain inequality (J.9).


appendixk.tex
\centerline{\bf Appendix K:  Observations concerning a Dedekind xi
function}

We note here some relations concerning a Dedekind xi function $\xi_k$,
extending some of our earlier derivative results for the Riemann xi function 
\cite{coffey}.

We let $k$ be an imaginary quadratic field of discriminant $d$, and 
$$\Phi(t)={\pi \over \sqrt{|d|}}\sum_F\sum_{m,n=-\infty}^\infty F(m,n)
\left[{{\pi t} \over \sqrt{|d|}} F(m,n)-1 \right ]\exp\left[-{{2\pi t}
\over \sqrt{|d|}}F(m,n)\right ], \eqno(K.1)$$
where the first sum is over the inequivalent classes of positive definite
integral quadratic forms of discriminant $d$.  Then Li \cite{li2001} has
very recently shown that
$$\xi_k(s)={4 \over w}\int_1^\infty \Phi(t)(t^s + t^{1-s})dt, \eqno(K.2)$$
for all complex $s$, where $w$ is the number of roots of unity.  
The function $\xi_k$ is entire and satisfies $\xi_k(s)=\xi_k(1-s)$
and $\xi_k(0)=\xi_k(1)=2^{r_1}hR/w$, where the number of real places $r_1=0$, 
the number of complex places $r_2=1$, $R=1$ is the regulator, and $h$ is the 
number of ideal classes of $k$.
%to verify R, etc.--look in Hua, Lang, and Weils' number theory books ...
%see also a 60-page paper of J. C. Lagarias and TBD--on dble zeta funcs?

The inequivalent classes of positive definite integral quadratic forms of
discriminant $d$ consist of the classes represented by the forms 
\cite{hua,li2001} $F(x,y)$ satisfying $b^2-4ac=d$ for either 
$-a<b\leq a<c$ or $0\leq b\leq a=c$.  Now it has also been shown 
\cite{li2001} that $F(x,y)=ax^2+bxy+cy^2$ satisfying this condition is
such that $F(m,n) \geq \sqrt{|d|}/2$ for all integers $m$, $n$ with 
$n \neq 0$.  It follows trivially that $\pi tF(m,n)/\sqrt{|d|}-1\geq
\pi t/2-1>0$ for all $t \in [1,\infty)$.  We then have the
immediate
\newline{\bf Proposition.}
For all the inequivalent classes of positive definite integral quadratic
forms of discriminant $d$, we have $\xi_k(s) >0$ for all real $s$.  
Furthermore, the integer-order derivatives
$$\xi_k^{(m)}(s)={4 \over w}\int_1^\infty \Phi(t)[t^s + (-1)^m t^{1-s}]
\ln^m tdt, \eqno(K.3)$$
satisfy $\xi_k^{(m)}(s) \geq 0$ for all $s \geq 1/2$.  The even order
derivatives obey the condition $\xi_k^{(2m)}(s) > 0$ for all $s \geq 1/2$.
Of course, as also seen by the functional equation for $\xi_k(s)$, the
odd-order derivatives $\xi_k^{(2m+1)}$ vanish at $s=1/2$; we have
$\xi_k^{(m)}(s)=(-1)^m\xi_k^{(m)}(1-s)$.  This Proposition seems to mean  
that $\xi_k(s)$ has no zeros for real values of $s$.  %is this correct?
%seem to need to check with Li and/or others ... What is the 'catch' here?

This Proposition extends some of the results of Ref. \cite{coffey}.
In turn, we may apply some of the explicit integration results obtained
there in order to evaluate the function
$$\gamma(a) \equiv \int_1^\infty \omega(t){{dt} \over \sqrt{t}}+    
\int_1^{\sqrt{|d|}/2a} \omega(t) {{dt} \over t}, \eqno(K.4)$$
introduced in Ref. \cite{li2001}, where $\omega$ is the $\theta$ series
given by $\omega(t)=\sum_{n=1}^\infty \exp(-\pi n^2t)$.  We have for the
first term on the right side of Eq. (K.4)
$$\sum_{n=1}^\infty \int_1^\infty e^{-\pi n^2 t}t^{-1/2} dt
={1 \over \sqrt{\pi}}\sum_{n=1}^\infty {1 \over n}\Gamma(1/2,n^2\pi)
=\sum_{n=1}^\infty {1 \over n}\left [1-Erf(n\sqrt{\pi})\right ],
\eqno(K.5)$$
where $\Gamma(x,y)$ is the incomplete Gamma function \cite{grad} and
$\Gamma(1/2,n^2 \pi)=\sqrt{\pi}[1-2n ~_1F_1(1/2;3/2;-\pi n^2)]$ and
$~_1F_1$ is the confluent hypergeometric function \cite{grad,pfqref},
such that $_1F_1(1/2;3/2;x)=(\sqrt{\pi}/2)Erf(\sqrt{-x})/\sqrt{-x}$, where
$Erf$ is the error function (probability integral) \cite{grad}.

For the second term on the right side of Eq. (K.4) we have
$$\sum_{n=1}^\infty \int_1^{\sqrt{|d|}/2a} e^{-\pi n^2t}t^{-1} dt
=\sum_{n=1}^\infty \left[Ei\left(-{{\pi n^2\sqrt{|d|}} \over {2a}}\right)
-Ei(-\pi n^2)\right ], \eqno(K.6)$$
where $Ei$ is the exponential integral \cite{grad}.  We also have various
elementary relations, including
$$\int_1^{\sqrt{|d|}/2a} e^{-\pi n^2t}t^{-1} dt=\pi n^2
\int_1^{\sqrt{|d|}/2a} e^{-\pi n^2t}\ln t dt+\exp\left(-\pi n^2\sqrt{|d|}/2a
\right) \ln\left({\sqrt{|d|} \over {2a}}\right ), \eqno(K.7)$$
obtained by integration by parts, and  
$$\int_1^{\sqrt{|d|}/2a} e^{-\pi n^2t}\ln t dt={1 \over {\pi n^2}}
\int_{\pi n^2}^{\pi n^2\sqrt{|d|}/2a} e^{-u}[\ln u - \ln (\pi n^2)] dt$$
$$={1 \over {\pi n^2}}\left[\int_{\pi n^2}^{\pi n^2\sqrt{|d|}/2a} e^{-u}\ln u 
du -\left(\exp(-\pi n^2\sqrt{|d|}/2a)-\exp(-\pi n^2)\right )\ln (\pi n^2) 
\right]. \eqno(K.8)$$

Moreover, the particular derivative values
$$\xi_k^{(2m)}\left({1 \over 2}\right )={8 \over w}\int_1^\infty \Phi(t)
t^{1/2} \ln^{2m} t ~dt, \eqno(K.9)$$
and
$$\xi_k^{(m)}(1)={4 \over w}\int_1^\infty \Phi(t)[t + (-1)^m]
\ln^m t ~dt, \eqno(K.10)$$
can be evaluated in terms of infinite series with the analytic methods of 
Ref. \cite{coffey}, and we note that $\xi_k^{(m)}(1) > 0$ for all nonnegative 
integers $m$.  These special values of Eq. (K.10) enter the particular 
logarithmic derivatives of Eq. (E.31) for $\lambda_n$.  
%Finally, we note
%that for the Dedekind zeta function of $k$, it appears that
%$\zeta_k^{(r_1+r_2-1)}(0) = -hR/w$, or, for the special case of this Appendix,
%$\zeta_k'(0) = -h/w$.  %is r_2=2 here ?? ... to examine yet ...
%see Intro section of a paper by Tom Weston of the Univ. of Michigan ...
%can I show this last relation directly by calculating the integrals
%involved?  Is perhaps a nice TO DO ........................................
%and I should then have $\zeta_k(0)=0$; also can calculate directly to show??








appendixl.tex
\centerline{\bf Appendix L:  Euler-Maclaurin summation applied to $S_1(n)$}

Here we apply Euler-Maclaurin summation to the form of the sum $S_1(n)$ 
given in Eq. (17) of the text.  Accordingly, we define the summand function
$$f(k) \equiv {n \over {2k+1}}-1+{{2^n k^n} \over {(2k+1)^n}}, 
~~~~~~k \geq 0, ~~~~ n \geq 2, \eqno(L.1)$$
such that $f(0)=n-1$ and $f(\infty)=0$.  We can write the arbitrary integer
order derivative of each term of Eq. (L.1).  For the first term on the right
side we have
$$\left({d \over {dk}}\right )^j{n \over {(2k+1)}}=(-1)^j {{2^jj!} \over
{(2k+1)^{j+1}}}n, ~~~~~~ j \geq 1.  \eqno(L.2)$$
When evaluated at $k=0$, this term gives $(-1)^j2^j j! n$.
In regard to the last term of Eq. (L.1) we have
$$\left({d \over {dk}}\right )^j{1 \over {(2k+1)^n}}=(-1)^j {{2^j(n)_j} \over
{(2k+1)^{n+j}}}, ~~~~~~ j \geq 1,  \eqno(L.3a)$$
where $(.)_j$ is the Pochhammer symbol, and
$$\left({d \over {dk}}\right )^r k^n={{n!} \over {(n-r)!}}k^{n-r}, \eqno(L.3b)$$
which can be equally expressed as
$$\left({d \over {dk}}\right )^r k^n=r!{n \choose r}k^{n-r} = (-1)^r (-n)_r 
k^{n-r}.  \eqno(L.3c)$$
Therefore we can write
$$2^n \left({d \over {dk}}\right )^\ell k^n(2k+1)^{-n}=2^n\sum_{m=n}^\ell
{\ell \choose m}{{(-1)^{\ell-m}2^{\ell-m}(n)_{\ell-m}} \over {(2k+1)^{n+\ell
-m}}} {{n!} \over {(n-m)!}} k^{n-m}.  \eqno(L.4)$$

A version of the Euler-Maclaurin formula, given that all derivatives of $f$
vanish at infinity, is
$$\sum_{n=M}^\infty f(n)=\int_M^\infty f(x) dx-\sum_{m=1}^\infty {B_m \over
{m!}}f^{(m-1)}(M), \eqno(L.5a)$$
where $B_m$ are Bernoulli numbers, or
$$\sum_{n=0}^\infty f(n) = \int_0^\infty f(x) dx + {1 \over 2}f(0)
-\sum_{m=2,even}^\infty {B_m \over {m!}}f^{(m-1)}(0). \eqno(L.5b)$$
By using Eq. (24) of the text for the integral in Eq. (L.5b) we therefore
obtain
$$S_1(n)={n \over 2}[\psi(n)+\gamma]
-\sum_{m=2,even}^\infty {B_m \over {m!}}f^{(m-1)}(0), \eqno(L.6)$$
where $\psi$ is the digamma function and $\gamma$ is the Euler constant.
The sums in Eqs. (L.5)-(L.6) are meant in an asymptotic sense; they are
highly unlikely to be convergent.  From Eq. (L.6) we may obtain the
successive approximations
$$S_1(n)={n \over 2}[\psi(n)+\gamma]+{n \over 6}
-\sum_{m=4,even}^\infty {B_m \over {m!}}f^{(m-1)}(0), ~~~~n > 1, \eqno(L.7a)$$
and
$$S_1(n)={n \over 2}[\psi(n)+\gamma]+{n \over {10}}
-\sum_{m=6,even}^\infty {B_m \over {m!}}f^{(m-1)}(0), ~~~~n > 3. \eqno(L.7b)$$
Equation (L.7a) is expected to be a useful approximate upper bound to
$S_1$ and Eq. (L.7b) an approximate lower bound for this sum.
When $n$ is sufficiently large, only the first term on the right side of
Eq. (L.1) contributes in Eqs. (L.6)-(L.7), giving $B_mf^{(m-1)}(0)/m!=-2^{m-1} 
B_m(n/m)$, as $m-1$ is always an odd integer.  Given alternation in sign
from $B_{2m}$ to $B_{2m+2}$, these successive terms will change sign also.
However, we emphasize that the sums in Eq. (L.6)-(L.7) are generally 
divergent.
