12:36 PM
(*start*)
Clear[f, A, B, x, n, k, s, rho];
f[x_] := Zeta[x];
A[n_, s_] :=
Sum[(-1)^(k - 1)*Binomial[n - 1, k - 1]/f[s + k/n - 1/n], {k, 1, n}]
B[n_, s_] :=
Sum[(-1)^(k - 1)*Binomial[n - 1, k - 1]/f[s + k/n], {k, 1, n}]
n = 30;
s = 14*I;
rho = s + 1/(1 - A[n, s]/B[n, s]);
N[%, n]
rho = -s + 1/(1 - B[n, s]/A[n, s]);
N[%, n]
Clear[f, A, B, x, n, k, s, rho];
Reduce[rho == s + 1/(1 - A/B) &&
s + 1/(1 - A/B) == Conjugate[-s + 1/(1 - B/A)], Re[rho], Complexes]
(*end*)
Re[rho] == 1/2 in the output is what I find interesting

2 hours later…
2:43 PM
I therefore believe that the Riemann hypothesis is equivalent to showing:
\lim_{n\to \infty } \, \left(\frac{1}{1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s-\frac{1}{n}\right)}}{\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{\zeta \left(\frac{k}{n}+s\right)}}}+s\right)=\lim_{n\to \infty } \, \left(\frac{1}{1-\frac{\sum _{k=1}^n \frac{(-1)^{k-1} \left( \begin{array}{c} n-1 \\ k-1 \end{array} \right)}{\zeta \left(\frac{k}{n}+s\right)}}{\sum _{k=1}^n \frac{(-1)^{k-1} \left( \begin{array}{c} n-1 \\ k-1 \end{array} \right)}{\zeta \left(\frac{k}{n}+s-