All Questions
11 questions
38
votes
2
answers
13k
views
What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
23
votes
1
answer
3k
views
More mysteries about the zeros of the Riemann zeta function
Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$.
Update on 1/5/2020: I added the section "more interesting ...
- 3,098
17
votes
2
answers
2k
views
Is this equivalent to RH - Riemann hypothesis?
$$\pi = 3\prod_{\zeta(1/2+it) = 0}\frac{9+4t^2}{1+4t^2}\iff\text{RH is true}.$$
- 1,305
14
votes
1
answer
1k
views
The location of the zeros of the "new" function $\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$
Defining the Psi function as $$\Psi(s)=\sum _{n=1}^{\infty }{\frac{1}{n!^s}}$$ and by studying, from a numerical point of view, the location of the zeros in the complex plane up to $0<\Im(s)<...
- 161
9
votes
0
answers
414
views
From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
- 3,098
4
votes
1
answer
928
views
On a possible equivalent of Riemann hypothesis
I've read in a Bombieri's paper on official problem statement of Riemann hypothesis for Clay Math institute's millennium problems, a statement and what I understood of it is the following :
The ...
- 790
4
votes
1
answer
333
views
Double sum over zeros of Riemann zeta-function
In a paper by Saffari and Vaughan there appears a complicated-looking double sum
$$\Sigma_1=\sum_{\rho_1}\sum_{\rho_2}\frac{(1+\theta)^{\rho_1}-1}{\rho_1}\cdot \frac{(1+\theta)^{\bar{\rho_2}}-1}{\bar{\...
- 465
3
votes
3
answers
366
views
zeros of a complex function defined by integers
This is a crosspost from Math.SE.
Is there a known increasing sequence of positive integers $\{\textbf{a}\} = a_0<a_1<a_2<.....$ such that all the zeros $z_k$ on $\Re[z]>0$ of the ...
- 179
3
votes
1
answer
309
views
Zeros of the derivative of $\xi$
In his paper on Zeros of derivatives of Riemann $\xi$-function on critical line Brian Conrey mention that
It can be shown that the Riemann hypothesis implies that all zeros of $\xi (s)$, the ...
- 61
2
votes
1
answer
2k
views
Books on complex analysis for self learning that includes the Riemann zeta function?
I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following:
Analytic number theory : the connection between complex analysis and
...
- 29
1
vote
0
answers
113
views
Are there any known statistics on the sign of the Stieltjes Constants?
The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
- 2,689