All Questions
34 questions
43
votes
3
answers
3k
views
Is this integral representation of $\zeta(2n+1)$ known?
Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...
- 383
29
votes
4
answers
5k
views
What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?
The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
- 2,200
15
votes
5
answers
2k
views
Zeros of the derivative of Riemann's $\xi$-function
The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...
- 11.1k
13
votes
2
answers
726
views
Special values of $\zeta$ outside the real line and the critical strip
The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
- 17.6k
11
votes
4
answers
707
views
Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?
Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
- 20.2k
10
votes
1
answer
731
views
What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?
A fair bit is known about rational zeta series. This includes identities like $$ \sum_{n=2}^{\infty} [\zeta(n) -1] = 1 . $$
Many more identities can be found in articles by e.g. Borwein and Adamchik &...
- 4,829
9
votes
2
answers
2k
views
References on Taylor series expansion of Riemann xi function
I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
- 603
8
votes
2
answers
5k
views
Inverse of the Riemann zeta function [closed]
I'm wondering if there is any information on the inverse of the Riemann zeta function (not it's reciprocal, but its functional inverse). This would obviously be a multi-valued function.
- 757
8
votes
2
answers
2k
views
Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$
Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
- 11.1k
7
votes
2
answers
788
views
Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$
Denote by $\Lambda(n)$ the von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If I recall ...
- 1,019
7
votes
1
answer
1k
views
The Correlation of the Möbius Function and Dirichlet Characters
Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$
In other words
$$\phi_{\chi}(n)=\sum_{d|...
- 11.4k
7
votes
2
answers
906
views
Positivity of the coefficients of Taylor series associated to the Riemann hypothesis
The question below relates to the paper "Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier. I'm asking it here because I am sure the ...
- 7,067
6
votes
1
answer
546
views
On Cramér's theorem about roots of Zeta function
Cramér proved the following theorem (see the announcement in [1] and [2]):
Consider the following function:
$$V(z)=\sum_k e^{\rho_kz}$$
Where $\rho_k$ runs through non trivial zeta zeros with $Im(\...
- 790
6
votes
0
answers
200
views
Empirical bounds on $\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right|$
It is reasonable to expect that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| < 2 \log \log t$$
for all $t\geq 4$ (say): a somewhat stronger bound is known for $t\geq 10^{165}$ or so (Theorem 5 ...
- 20.2k
5
votes
1
answer
423
views
A generating function for non-trivial zeros of Riemann zeta function
Suppose $0^+_\zeta$ is the set of non-trivial zeros of the Riemann zeta function $\zeta(s)$ which lie on or to the right of the critical line and above the $x$-axis, i.e,
$$0^+_\zeta = \{s \in \mathbb{...
- 51
5
votes
1
answer
273
views
Where can I find this result of Ingham?
Sometime ago, I read somewhere (should be in Titschmarsh) that, if $N(\sigma, T)$ denotes the number of zeros of the Riemann zeta function $\zeta(s)$ with $\Re(s)\geq \sigma>1/2$ up to height $T>...
- 1,019
5
votes
0
answers
322
views
Approximating $\zeta^{(r)}(s)$ by a sum
Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
- 20.2k
5
votes
0
answers
504
views
An explicit formula for $\zeta(2m+1)$ with good convergence
The question: Is the following formula known?
$$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m \frac{(2^{2v-2k+...
- 293
4
votes
1
answer
248
views
Are there any extensive treatments on rational zeta series?
I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...
- 4,829
4
votes
0
answers
168
views
Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 11.1k
4
votes
0
answers
280
views
Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?
Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
- 1,018
4
votes
0
answers
170
views
Can $ x \sum_{k=1}^{\infty} \frac{1}{k} \Big{(}- \gamma - \psi \big{(}1-\frac{x}{k} \big{)} \Big{)} $ be simplified?
I'm interested in sums of the form $$f_{p} (x) = \sum_{k=2}^{\infty} \zeta(k)^{p} x^{k} .$$
For $p=1$, the following result is known: $$f_{1} (x) = -x \big{(}\psi(1-x) + \gamma \big{)} .$$
(That is, ...
- 4,829
3
votes
1
answer
1k
views
Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of $...
- 534
3
votes
2
answers
386
views
Explicit bound on $\zeta(s)$ inside a zero-free region?
Does anybody know of a place in the literature where one can find an explicit result of the form $|\zeta(\sigma+it)|\leq C \log t$ for $t$ within a zero-free region (assuming $t$ is larger than an ...
- 20.2k
3
votes
1
answer
330
views
Turan Inequalities
A real entire function
$$\psi(x)=\sum_{k=0}^{\infty} \gamma_k\frac{x^k}{k!}$$
is said to be in the Laguerre-Polya class, denoted $\psi(x) \in \mathcal{LP}$, if it can be represented in the form
\...
- 73
3
votes
2
answers
316
views
Explicit formula: explicit work with general smoothing?
The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...
- 20.2k
3
votes
0
answers
159
views
Is there a general relationship between definite integrals over functions involving the complete elliptic integral of the first kind and zeta values?
Background
Let $\textbf{K}(k)$ be the complete elliptic integral of the first kind, where $k$ is its elliptic modulus [1]. Moreover, define $k' := \sqrt{1-k^{2}} $ as its complementary modulus. I've ...
- 4,829
3
votes
0
answers
219
views
What is known about products of zeta values?
A couple of years ago, I asked this MSE question on the evaluation of the product of even zeta values: $$ \prod_{n=1}^\infty \zeta(2n) \approx 1.82 \quad .$$ While it can be shown that the product ...
- 4,829
2
votes
2
answers
363
views
Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
user536681
1
vote
2
answers
852
views
Trying to debunk a claim
Claim: Take any function $f(t) > 0$ for $t > 0$, such that $f(t) \to \infty$ as $t \to \infty$, then for $\sigma > 0$ $$|\zeta(\sigma + it)| = o(f(t))$$
Is there any already existing ...
- 731
1
vote
1
answer
190
views
Values of the completed Riemann $\xi(1+it)$ for small t?
I'm editing this question heavily for clarity:
I am looking for methods to compute $\zeta(1+it)$, or the (partially) completed Riemann zeta function
$$\pi^{-s/2}\Gamma(s/2)\zeta(s)$$
along the line ...
- 3,799
1
vote
0
answers
253
views
Who formulated the conjecture that the set of real parts of zeros of the Riemann zeta function is dense in $[0,1]$?
Does anyone know who formulated this conjecture related to Riemann's zeta function?
Conjecture. The set $$\{ x : \exists y \space \space \zeta (x+iy) = 0\}$$ is dense in $[0, 1]$.
In ...
0
votes
1
answer
195
views
Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?
Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$.
Also $\lambda_n$ is given as a sum over the non ...
user469908
0
votes
1
answer
186
views
On the explicit upper bound of $|\log\zeta(s)|$ near $\Re(s)=1$
I learn from Montgomery & Vaughan's Multiplicative Number Theory I: Classical theory that there exists $c_1,c_2>0$ such that whenever $\sigma\ge1-c_1/\log|t|,|t|\ge4$ there is
$$
|\log\zeta(\...
- 1,315