3
votes
0answers
60 views

How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
3
votes
0answers
62 views

Can you prove that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$, assuming LeClaire's approximation?

Can you prove using double series reversion that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$ (as their convergent), with initial guess for the real part $r$ to be ...
1
vote
0answers
62 views

Zeta zeros by recurrence of zeta function, but this is useless isn't it?

One more useless question of mine can't do this site any harm. So here we go. The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting ...
3
votes
1answer
67 views

A double sum and its relation to a simple sum, is this an identity for any complex number $S=a+i b$ and any integers n and t?

Does: $$\sum _{m=1}^t \lim_{s\to \text{S}} \, \zeta (s) \sum _{k=(m-1) n+1}^{m n} \frac{1-\text{If}[k \bmod n=0,n,0]}{k^{s-1}}$$ equal: $$\lim_{s\to \text{S}} \, \zeta (s) \sum _{k=1}^{n t} ...
11
votes
2answers
2k views

Show how to calculate the Riemann zeta function for the first non-trivial zero

I have very little understanding on how complex functions work but was wondering if someone could show what the summation of the zeta function simplifies to when $s$ is the first non-trivial zero of ...
0
votes
0answers
69 views

Zeroes of $s+\sum\limits_{n=2}^\infty \frac{(-1)^{n+1}}{n^s\ln n} $?

Where are the solutions of the equations $$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$ Since the ...
4
votes
2answers
108 views

Are the solutions to $1+1/2^s+1/3^s=0$ known?

For $s$ a complex number, are the solutions to this equation known? $$1+1/2^s+1/3^s=0$$ Borwein et alia have studied the partial sums of the zeta function: Zeros of partial sums of the Riemann ...
3
votes
1answer
134 views

Convergence of the Zeta and Phi functions

I want to show that the following functions (in the picture) are absolutely and locally uniformly convergent if real part of complex number $s$ is bigger than 1. Absolute part for zeta function is ...
6
votes
1answer
643 views

What is the formula for the first Riemann zeta zero?

I found this approximation of which an earlier version I posted in the chat room: $$7 \pi -\text{Log}\left[\frac{7}{2} e^{-7 \pi /2}+\frac{5}{2} e^{-5 \pi /2}+\frac{3}{2} e^{-3 \pi /2}+e^{5 \pi /2}+2 ...