Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with riemann-zeta-function
Search options questions only
not deleted
user 12481
The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.
8
votes
2
answers
757
views
Are there known non-real zeros of derivatives of Riemann zeta with 0 < Re(s) < 1/2?
According to New zero free regions for the derivatives of the Riemann zeta function
assuming the Riemann Hypothesis, $\zeta^{(k)}(s)$ has
at most a finite number of non-real zeros with $\operatorname{ …
- 25.4k
3
votes
1
answer
414
views
How is "large" defined in an equality for the modulus of Riemann zeta?
This paper p.4 claims:
Corollary C. Assume RH. For all large $t$ we have
$$|\zeta(\frac12 +it)| \le \exp\left(\frac38 \frac{\log{t}}{\log{\log{t}}}\right) \qquad (1) $$
$t$ a Gram points often appe …
- 25.4k
4
votes
0
answers
185
views
Nearest integers to derivatives of zeta
Closely related, but different from this solved quesion
Let $\zeta^{(k)}(s)$
denote the $k$-th derivative of Riemann zeta function.
For real $x$, let $[x]$ denote the nearest integer to $x$.
Conjectur …
- 25.4k
2
votes
2
answers
547
views
What are the fallacies that this RH inequality may fail at most finitely often?
According to "EQUIVALENCES TO THE RIEMANN HYPOTHESIS
p.4
Let $g(n)$ be the maximal order of a permutation of n objects
RH Equivalence 3.3. The Riemann Hypothesis is equivalent to
$\log{g(n)} < Li^{ …
- 25.4k
19
votes
4
answers
3k
views
zeta(3) in terms of derivatives of zeta at 1/2 and pi
Got numerical support that for odd $n$, $\zeta(n)$ might be
expressed in terms of the derivatives of $\zeta(\frac12)$.
Based on More Zeta Functions for the Riemann Zeros, Andre Voros, p.12, Table 3:
…
- 25.4k
7
votes
0
answers
421
views
$\zeta(x)$ in terms of $\zeta'(x),\zeta'(1-x),\Gamma,\psi$
By differentiating $\xi$ and solving for $\zeta(1-x)$:
$$ \zeta(1-x) = \frac{2(\zeta'(x)\Gamma(x/2)+\Gamma((1-x)/2) \zeta'(1-x)\pi^{x-1/2}) )}{\Gamma((1-x)/2) \pi^{-1/2+x}(2\log\pi -\psi((1-x)/2)-\p …
- 25.4k
8
votes
1
answer
623
views
Is there always a zero between consecutive local extrema of $\Re \zeta(1/2+it)$ (or $\Im \ze...
Based on limited numerical evidence, I am inclined to suspect that
there is always zero of $\Re \zeta(1/2+it)$ between consecutive local
extrema of $\Re \zeta(1/2+it)$
(and the same for $\Im \zeta(1/2 …
- 25.4k
3
votes
2
answers
464
views
On the critical line $ \Re \zeta'(s)/\zeta(s) =? 1/2 \log(\pi) - 1/2 \Re \psi(s/2)$ ?
For $\Re s = 1/2$ numerical evidence suggest:
$$ \Re \zeta'(s)/\zeta(s) = 1/2 \log(\pi) - 1/2 \Re \psi(s/2) \qquad (1) $$
How this was found. Consider the symmetrized zeta function
$\zeta^*(x)= \pi^{- …
- 25.4k
6
votes
2
answers
691
views
Bounds for $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{(k-1)!\zeta(2k)}$
Let $$ f(x) = \sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{(k-1)!\zeta(2k)}$$
Are there lower bounds, upper bounds or (unlikely) simpler closed form
for $f(x)$?
The bounds for Bernoulli numbers I fou …
- 25.4k
4
votes
1
answer
1k
views
zeta(2k+1) is a rational multiple of pi^{2k} zeta'(-2 k) ?
Probably this is well know and elementary and will delete it, but couldn't find it on the web.
Got a sketch of proof and numerical evidence that
$\zeta(2k+1)$ is a rational multiple of $\pi^{2k} \ze …
- 25.4k
2
votes
0
answers
363
views
Complex zeros of $\zeta'(s)/\zeta(s) + \zeta'(1-s)/\zeta(1-s) $ = simpler expression (except...
Let $ G(s) := \frac{\zeta'(s)}{\zeta(s)} + \frac{\zeta'(1-s)}{\zeta(1-s)}$
where $s$ is not a zero of zeta.
$G$ has real zeros and a pair of complex zeros near $\frac12 \pm 6i$.
Partial results:
By …
- 25.4k
3
votes
2
answers
495
views
Finite sum seemingly related to nontrivial zeta zeros
For $t \in \mathbb{R}$ define
$$ F(t) = \sum_{n=1}^{[t]} \frac{(-1)^{(n-1)}}{n^{\frac12 + it}}$$
Let $\operatorname{Arg}(t)$ be $\operatorname{atan2}(\Im t , \Re t)$ -
basically this is $\arctan$, but …
- 25.4k
1
vote
0
answers
132
views
Conjectured alternate form for vanishing of $\Re\zeta(1/2+it)$ except at zeros
Heavily based on Agno's question.
Define:
$$ \chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2}) $$
Agno conjectured: only for $\sigma=\frac12$, $\Re(\chi(s)) = \Re(\zeta(s)) =0$ is always true, except …
- 25.4k
7
votes
0
answers
156
views
How comes vanishing of the real part of function involving zeta is very well approximated by...
In this question
Agno asked about the zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$.
I fixed $a=2$ and the minus sign and defined:
$$
f(s)=\Re \left( \zeta\left(\frac{ …
- 25.4k
4
votes
1
answer
464
views
Is this differential equation for zeta on the critical line? One can compute it from its der...
Looks like on the critical line one can compute
$\zeta(1/2+it)$ from $\zeta^{'}(1/2+it)$ and simpler functions.
Let
$$
\begin{aligned}
f(t)= & 2\, \left( {\frac { \left( \left| \zeta^{'} \left( 1/ …
- 25.4k