TITLE: If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$?
QUESTION [15 upvotes]: Helmut Hasse has proved that for $s \in \mathbb{C}-\{1\}$ the Riemann zeta function can be written as:
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k}\ \frac{1}{(k+1)^{s}}$$
Let $K(a,b) = \frac{\gcd(a,b)}{a+b}$. Then we might define:
$$\hat{\zeta}(s,a)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k} K(k,a)^{s}$$
Then $\hat{\zeta}(s,1) = \zeta(s)$.
I looked at some non-trivial zeros of $\zeta$ and computed numerically in SAGEMATH:
If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$ or is this just some bug / numerical curiosity in my code?
Here is the SAGE Code:
def K(a,b):
    return gcd(a,b)/(a+b)

def zzeta(s,a=1,N=400):
        return 1/(1-2**(1-s))*sum([ 1/2**(n+1)*sum([ (-1)**k*binomial(n,k)*K(k,a)**s for k in range(n+1)]) for n in range(0,N+1)])


zz = zeta_zeros()

for z in zz[0:100]:
    print( abs(zzeta(1/2+z*I,3))**2)

And here is some of the ouptut:
0.5000000001016516
0.49999999980871757
0.49999999990048116
0.4999999999150287
0.5000000001425052
0.5000000003259728
0.4999999999688171
0.5000000000737418
0.5000000004006991
0.5000000003452242
0.49999999982210547
0.5000000000784613
0.49999999939079287
0.5000000012082603
0.5000000000591772
0.5000000002545129
0.4999999996870637

Some data for $a=2$:
0.5027599264329976
0.4120370789607451
0.3869873346989051
0.2811880366841188
0.2594539123560252
0.7742279955909737
0.18310290214474273
0.4006464279725155
0.2723910071065707
0.16780149194046892

REPLY [5 votes]: The intent of this post is to share some results as requested in the comments above and assumes the following definitions from the original question above.

(1) $\quad\zeta(s)=\underset{N\to\infty}{\text{lim}}\left(\frac{1}{1-2^{1-s}}\sum_\limits{n=0}^N\frac{1}{2^{n+1}}\sum_\limits{k=0}^n(-1)^k\left(
\begin{array}{c}
 n \\
 k \\
\end{array}
\right)\frac{1}{(k+1)^s}\right)$
(2) $\quad\hat{\zeta}(s,a)=\underset{N\to\infty}{\text{lim}}\left(\frac{1}{1-2^{1-s}}\sum_\limits{n=0}^N\frac{1}{2^{n+1}}\sum_\limits{k=0}^n(-1)^k\left(
\begin{array}{c}
 n \\
 k \\
\end{array}
\right)K(k,a)^{s}\right),\qquad K(a,b)=\frac{\gcd(a,b)}{a+b}$

The following figure illustrates formula (2) above for $|\hat{\zeta}(s,3)|$ plotted along the critical line $s=\frac{1}{2}+i\ t$. The red discrete portion of the plot represents the evaluation of formula (2) for $|\hat{\zeta}(s,3)|$ at the first 10 non-trivial zeta zeros in the upper half plane. In the plot in the figure below formula (2) was evaluated  using an upper limit of $N=400$.


Figure (1): Illustration of formula (2) for $|\hat{\zeta}(s,3)|$ evaluated along the critical line

The following conjectured formulas based on my Questions related to globally convergent formulas for the Dirichlet eta function $\eta(s)$ seem to evaluate considerably faster than formulas (1) and (2) above and perhaps provide some additional insight.

(3) $\quad\zeta(s)=\underset{N\to\infty}{\text{lim}}\left(\frac{1}{\left(1-2^{1-s}\right)\ 2^{N+1}}\sum\limits_{n=0}^N (-1)^n\ \frac{1}{(n+1)^s}\sum\limits_{k=0}^{N-n} \left(
\begin{array}{c}
 N+1 \\
 N-n-k \\
\end{array}
\right)\right)$
(4) $\quad\hat{\zeta}(s,a)=\underset{N\to\infty}{\text{lim}}\left(\frac{1}{\left(1-2^{1-s}\right)\ 2^{N+1}}\sum\limits_{n=0}^N (-1)^n\ K(n,a)^{s}\sum\limits_{k=0}^{N-n} \left(
\begin{array}{c}
 N+1 \\
 N-n-k \\
\end{array}
\right)\right)$

The following figure illustrates formula (4) above for $|\hat{\zeta}(s,3)|$ plotted along the critical line $s=\frac{1}{2}+i\ t$. The red discrete portion of the plot represents the evaluation of formula (4) for $|\hat{\zeta}(s,3)|$ at the first 10 non-trivial zeta zeros in the upper half plane. In the plot in the figure below, formula (4) was evaluated using an upper limit of $N=400$.


Figure (2): Illustration of formula (4) for $|\hat{\zeta}(s,3)|$ evaluated along the critical line

Formula (5) below is from the accepted answer to this question posted by Lucia.
(5) $\quad\hat{\zeta}(s,3)=\frac{2^{-s}-1}{1-2^{1-s}}+\zeta(s)+\frac{3^s-1}{3^s} \zeta(s)$

The following figure illustrates formula (5) above for $|\hat{\zeta}(s,3)|$ and $\left|\frac{2^{-s}-1}{1-2^{1-s}}\right|$ in blue and orange respectively where both functions are evaluated along the critical line $s=\frac{1}{2}+i\ t$. The red discrete portion of the plot represents the evaluation of formula (5) for $|\hat{\zeta}(s,3)|$ at the first 10 non-trivial zeta zeros in the upper half plane.

Figure (3): Illustration of formula (5) for $|\hat{\zeta}(s,3)|$ evaluated along the critical line