Questions tagged [riemann-zeta-function]

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.

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Bounds for analytic circles

It is known that for certain particular entire functions $f(s)$ of first order, in the circle $|s| = p$, if $\epsilon$ is a positive number as small as desired, the following bound holds: $$|f(s)| = O(...
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Lindelöf hypotheses for derivatives of zeta

The Lindelöf hypothesis says that if we have: $$\zeta(\sigma+iT)=\mathcal O(T^a)$$ Then if one considers $\sigma=1/2$ then $\inf a=0$. Further, from convexity and the functional equation this implies ...
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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 ...
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On the asymptotic behaviour of the series $\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$ near $s=0$

I am interested in determining the behaviour of the the series/function $$f(s)=\sum_{n=1}^{\infty} \left ( \frac{\zeta(ns)}{n}+\frac{s}{1-ns}\right )$$ near $s=0$. It is clear that $f(0)$ is undefined....
Tian Vlašić's user avatar
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When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...
free_lancer's user avatar
3 votes
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Generalized harmonic numbers and Riemann zeta function

The $n$-th harmonic number is defined as $$ H_{n}=\sum\limits_{k=1}^{n}\frac{1}{k}, $$ and the generalized harmonic numbers are defined by $$ H_{n}^{(m)}=\sum\limits_{k=1}^{n}\frac{1}{k^m}. $$ It is ...
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The logarithmic derivative of a twisted L-function?

Let $F$ be a quadratic number field with class number $h_F = 1$. Let $\zeta_F$ be the Dedekind zeta function, we have $$ \frac{\zeta_F ' (1+it)}{\zeta_F (1+it)} \ll \frac{\log t}{\log\log t} .$$ (I ...
Misaka 16559's user avatar
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Real part of the Riemann zeta function

Consider the real part of the Riemann zeta function on the critical line. Are there any results for the number of zeros of this real function in the interval [0,T]?
Autovetor's user avatar
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A conjectured series expression for the Riemann $\xi$-function and/or Completed L-series. Could this be proven?

This post builds on an MSE question about a conjectured series expression for the Riemann $\xi$-function: $$\xi(s) = \xi(1-s) = \sum_{n=1}^\infty (-1)^{n+1}\,\big(\xi\left(s+in\right)+\xi\left(1-s+in\...
Agno's user avatar
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Approximating $\zeta'/\zeta$ (and its derivatives) by a finite sum

Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a ...
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Does this partial sum over primes spike at all zeta zeros?

Below is a plot of $\exp \sum _p^x -\frac{\cos \left(x \log \ p\right)}{\sqrt{p}}$, where $p$ runs over the primes, and the $x$-values of the Riemann $\zeta$ zeros are marked with dashed lines: Below ...
martin's user avatar
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Reference book on Riemann zeta function and random matrices

What is a reference book to understand the relation between the Riemann zeta function and random matrices?
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Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$

This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted. Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
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Are $\zeta'(0)$ and $\beta'(0)$ algebraic numbers?

Let $\zeta$ be the Riemann zeta function and $\beta$ the Dirichlet beta function. We know that $\zeta (0)=-1/2$ and $\beta (0)=1/2$ are algebraic numbers over $\mathbb{Q}$. This led me to the ...
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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 (...
H A Helfgott's user avatar
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Xi function representation

Would it be correct to write down the following, or is it completely wrong? $$ \Xi(z) = \frac{1}{2} \int_{-\infty}^{\infty} e^{-\pi x^2} \theta''(x) \sin(zx) \, dx, $$ with $$ \theta(x) = \sum_{n=-\...
Bo Jonsson's user avatar
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Spiralling cycles surrounding zeros

The following came up, as a vague idea, in dialogue with a bright, female, 20 year old student of mine. It is a bit vague, but it seems that conjecture 1 is not present in the literature, which seems ...
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Three conjectural series for $\pi^2$ and related identities

Recently, I found the following three (conjectural) identities for $\pi^2$: $$\sum_{k=1}^\infty\frac{145k^2-104k+18}{k^3(2k-1)\binom{2k}k\binom{3k}k^2}=\frac{\pi^2}3,\tag{1}$$ $$\sum_{k=1}^\infty\frac{...
Zhi-Wei Sun's user avatar
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2 votes
1 answer
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Grouping the zeros of imaginary parts derivatives of zeta on horizontal lines

Let $\zeta^{(k)}(s)$ denote the $k$-the derivative of zeta function. Let $S=\{\Im(\zeta(\sigma +i t)),\Im(-\zeta^{(1)}(\sigma + i t),\Im(\zeta^{(2)}(\sigma + i t))\}$ We are interested in the plots of ...
joro's user avatar
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The argument of Riemann zeta function and the number of zeros on the critical line

Back ground I studied the proof of "$KT$ zero theorem" and "$KT\log T$" theorem in Edwards book. And I'm looking for other kind of evaluation of the number of zeros on the line. ...
George's user avatar
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Question on the inverse Mellin transform $p(x)=\mathcal{M}_s^{-1}\left[-\xi(s)\,\frac{\zeta'(s)}{s\,\zeta(s)^2}\right]\left(\frac{1}{x}\right)$

Consider the function $$p(x)=\underset{K\to\infty}{\text{lim}}\left(\sum\limits_{k=1}^K \Lambda(k) \left(\frac{2 \pi k^2}{x^2}-1\right) e^{-\frac{\pi k^2}{x^2}}\right)\tag{1}$$ where $$P(s)=s\, \...
Steven Clark's user avatar
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An evaluation of the second Chebyshev function

Let $$ \begin{align} \Lambda (n) & &\text{the Von Mangoldt function,}\\ \psi(x)&:=\sum_{n=1}^{[x]}\Lambda (n)&\text{the econd Chebyshev function,}\\ T(x)&:=\sum_{n=1}^{[x]}\log(n). ...
George's user avatar
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1 answer
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Zero-free regions of $\zeta(s)$ equivalent to prime number theorems with error bound

A 1950 result of Tur'an establishes an equivalence between any prime number theorem of the form $\operatorname{li}(x)-\pi(x)= O(xe^{-C(\log x)^\alpha}) \ (x \to \infty)$ and a certain class of zero-...
Jesse Elliott's user avatar
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On gaps between consecutive zeros of the Riemann zeta function

Let $\gamma$ denote the imaginary part of a non-trivial zero of the Riemann zeta function. Do there exist some function $f$ such that $\gamma_{n+1} - \gamma_n > f(n)>0$ for all large $n$? To be ...
Higgs Boson's user avatar
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2 answers
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On the error term of the Riemann explicit formula

Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for $$f(x, T) := \...
Higgs Boson's user avatar
3 votes
1 answer
447 views

Prime number theorem via the explicit formula

Can the prime number theorem be obtained from the explicit formula, $\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}{\rho}+O(1)$? Here, $\psi(x)=\sum_{k=1}^\infty\sum_{p^k<x}\log p$
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Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?

The well-known integral expression for the entire function: $$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$ ...
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On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:

Consider the analytic function $g(x)$ Now define $f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$ Such that $|f(x+it)|=o(e^{2πt})$ uniformly for every $x$...
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1 answer
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On the upper bound for $|\zeta(s)|$ near the zeta zeros

Let $T \in \mathbb{R}$ be large and $\rho$ be a non-trivial zero of the Riemann zeta function. Assume that $|\rho|=|\rho_T| \approx T$ and let $\varepsilon_T \approx \frac{\log \log T}{\log T}$. Is it ...
Higgs Boson's user avatar
3 votes
1 answer
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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 ...
Tokita Ohma's user avatar
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0 answers
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Zeta function associated with a function $f$

Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define $$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt. $$ Is there a general formula that ...
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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 $\rho=\sigma+it$. ...
EGME's user avatar
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7 votes
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On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?

After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$. I. Recurrences involving $\zeta(5)$ In Cohen's 2022 paper, ...
Tito Piezas III's user avatar
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Residue calculation for Eulerian expansion of the cotangent

I am looking for ideas on proving the Eulerian expansion of the cotangent using residue calculation: $$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\left(\frac{1}{z+n}+\frac{1}{z-n}\right), \ z\in\...
L.L's user avatar
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Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$?

I originally asked this question on Math StackExchange a few months ago and no answers or even comments have yet been posted, so I'm asking this question again here on Math OverFlow. This Math ...
Steven Clark's user avatar
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An inequality related to Catalan's constant and $\zeta(3)$

Problem : Show that : $$\frac{1}{\zeta(3)}<2C-1$$ Where we can see the zeta function and the Catalan's constant . After a bounty on Maths Stack Exchange there is no satisfying answer . See https://...
DesmosTutu's user avatar
7 votes
2 answers
822 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 ...
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What's the meaning of this relation between volumes of $n$-balls and umbral calculus?

The volume of an $n$-ball of radius $1$ is $$V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$$ The functional equation of Riemann zeta function is $${\displaystyle \pi ^{-{s \...
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What heuristic arguments support Montgomery's conjecture for primes in short intervals?

I have a question regarding a conjecture due to H. L. Montgomery on the number of primes in short intervals. The conjecture apparently arises from probabilistic reasoning upon assuming the Riemann ...
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proving inequality in Riemann zeta function

Recently I have made some interesting observations on the limit $$\lim_{k\rightarrow \infty}{\sum_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}. $$ When this ...
MrPie 's user avatar
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Asymptotics of the Liouville sum at the primes

Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
user501735's user avatar
3 votes
2 answers
700 views

Do all rigorous proofs of Euler's product for $\zeta (s)$ use infinitude of primes?

There are two proofs of $$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ prime}}\frac{1}{1-p^{-s}},\quad \Re (s)\gt 1$$ which I'm aware of. I'll call the first one the Sieve proof and the second one ...
Vestoo's user avatar
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1 answer
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Derivative of the Riemann zeta function at $z=-2$

I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant ...
L.L's user avatar
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0 answers
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Reference for explicit formula used by Ramanujan

In his work on highly composite numbers http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf , Ramanujan used a version of an explicit formula (equation (329) on page 133) relating primes and zeros of ...
Dekimshita's user avatar
3 votes
1 answer
287 views

Derivative of zeta at positive even integers

Is there a general formula that sums up all values of $ζ′(2n)$, such that $n\in\mathbb{N}$?
L.L's user avatar
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1 answer
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Large values of $\zeta(1/2+it)$ from sums of short moments

In a now classical paper, Iwaniec proved the following theorem. Theorem. Let $T \geq 2$, $T^{1/2} < T_0 \leq T$ and $T \leq t_1 < t_2 < \cdots < t_R \leq 2T$, $t_{r+1} - t_r \geq T_0$. ...
Joshua Stucky's user avatar
10 votes
2 answers
896 views

$\psi(x)-x$ on average

This is a reference question: Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that $$ \int_2^x (\psi(y)...
Dr. Pi's user avatar
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1 vote
1 answer
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Best possible unconditional partial sum estimate of $\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$:

Consider the following partial sum: $$S(x,n)=\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$$ Here p runs through primes and $n$ is constant What is the best possible unconditional( using best known version ...
Zaza's user avatar
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0 answers
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What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?

Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by $$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$ (the nonvavishing of the denominator being a bit weaker than the prime number ...
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3 votes
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What is the behavior of the argument of Riemann’s Zeta function on other verticals inside the critical strip, apart from the critical line?

What is known about the behavior of the argument of Riemann’s Zeta function on other verticals inside the critical strip apart from the critical line ? Are there any omega type theorems in this case, ...
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