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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Is the proof in "On Hilbert’s 8th Problem" published on Brazilian Journal of Probability and Statistics correct? [closed]

The article can be freely accessed here. The proof is only five pages. I am quite in doubt. A new version (2021) of that paper can be found here.
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$\lim_{x\to \infty} \left(\sum_{n\leq x} (\log n)^k/n - \int_1^x (\log t)^k/t\right) = \text{?}$

It is easy to see (by Euler-Maclaurin, say, or just by thinking of a graph) that $$\lim_{x\to \infty} \sum_{n\leq x} \frac{(\log n)^k}{n} = \int_1^x \frac{(\log t)^k}{t} + C + O\left(\frac{(\log x)^k}{...
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Convergence of zeta Euler product with additional term

Let's consider the following Euler product ($s=\sigma+it)$: $$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$ So for $\sigma>1$, it is clear the product converges and we have: $$...
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On general 'explicit' expression for constant term in finite sum of function of primes

Consider the following finite sum $$\sum_{p\leq x}f(p) = S(x)+C$$ Here, $f(x)$ is smooth $p$ is prime $S(x)$(=smooth+oscillation) is also a 'function'; $C$ is a constant We also know the following $$\...
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Is $\lim_{s \rightarrow 1} E(f(X_s)) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N f(k)$?

Let $s>1$ be a real number. We look at the zeta probability function / Zipf probability function defined as: $$P(X = n) = \frac{1}{n^s \zeta(s)}$$ Suppose $f: \mathbb{N} \rightarrow \mathbb{R}$ is ...
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Heuristic model for Lehmer pairs?

Rodgers and Tao proved that the De Bruijn–Newman constant $\Lambda$ is non-negative. The study of $\Lambda$ goes back at least to Lehmer's paper, On the roots of the Riemann zeta-function, whose ...
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Odlyzko's reformulation of Montgomery's pair correlation conjecture

In his famous paper, On the distribution of spacings between zeros of the zeta function (https://www.ams.org/journals/mcom/1987-48-177/S0025-5718-1987-0866115-0/S0025-5718-1987-0866115-0.pdf), Odlyzko ...
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Evaluations of two new series involving Lucas $v$-sequences

Let $A$ and $B$ be integers. The Lucas $v$-sequence $v_n(A,B)\ (n=0,1,2,\ldots)$ is defined by $v_0(A,B)=2,\ v_1(A,B)=A$, and $$v_{n+1}(A,B)=Av_n(A,B)-Bv_{n-1}(A,B)\ \ \ (n=1,2,3,\ldots).$$ From the ...
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Proof for $\Phi(t)$ is strictly decreasing for $t>0$ in Riemann's zeta function

I am looking for reference for proof that $\Phi(t)$ is strictly decreasing for $t>0$ and the first derivative of $\Phi(t)$ is negative for $t>0$ (see Page 5 in Conrey's article below) Conrey ...
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$\zeta(s) = 1 + X^{1-s}/(s-1) + ...$?

Let $s = \sigma+ i t$ with $0\leq \sigma\leq 1$, $|t|\leq X$, where $1\leq X<2$. It is easy to use the Euler-Maclaurin formula to prove a result of the form $$\left|\zeta(s) - 1 - \frac{X^{1-s}}{s-...
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$\zeta(s) = \sum_{n\leq x} n^{-s} - x^{1-s}/(1-s) + ...$ through bounded-order Euler-Maclaurin?

It is a basic classical result (Titchmarsh Thm 4.11; credited to Hardy-Littlewood) that, uniformly for $\Re s \geq \sigma_0>0$, $t\leq 6 x$ (say), $$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} - \frac{...
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Integral of $|1/\zeta(\sigma+i T)|$ (or $|(1/\zeta(\sigma+i T))^{(k)}|$) on a horizontal half-line in the left upper quadrant

Let $T_0\geq 20$. Let $L$ be the half-line from $-\infty + i T$ to (say) $-1/2 + i T$. Since $|\zeta(s)|$ is roughly proportional to $(T/2 \pi e)^\sigma$ for $s=\sigma+ i T$ on $L$, it is clear that ...
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Correct growth rate of logarithmic derivative of zeta, outside critical strip

Let $\zeta$ be the Riemann zeta-function, and let $t> 0$. I'm interested in the growth rate of $$ \left|\frac{\zeta'}{\zeta}\left(-\frac{1}{2}+it\right)\right| $$ as $t\to\infty$. It is easy to ...
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Assuming Riemann's Hypothesis, is the argument of Zeta O(log T) in a neighborhood of the critical line?

It is known that: $$\frac1\pi\arg\zeta\left(\frac12+iT\right)=\mathcal O(\log T)$$ Question. Assuming  Riemann's Hypothesis,  is it true that  for a sufficiently small positive $\epsilon$ $$\frac1\pi\...
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Intuition for the bias of the partial sums of the Liouville function

It's a well known result that the Dirichlet series of the Liouville function $ \lambda(n) $ is given by $$ \sum_{k=1}^{\infty} \frac{\lambda(k)}{k^s} = \frac{\zeta(2s)}{\zeta(s)} $$ If we use Perron's ...
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The exploration of the asymptotic behavior of a simple sum. $\sum_{k=1}^{\infty} (k^{1/k} - 1)$ [closed]

An informal investigation of a sum. Consider this sum: $$S =\sum_{k=2}^{\infty}(k^{1/k} -1)$$ Does this converge? How does it behave as it diverges, if it diverges? If $k$ equaled $1$ we would get $0$ ...
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Ramanujan sums, zeta functions

We know that the Ramanujan sums $$c_k:=\sum_{h<k, \gcd(h,k)=1}\cos \frac{2h \pi}{k}$$ have the property that $$\sum_{n \ge 1} \frac{c_n}{n^s}=\zeta(s)^{-1},\quad \text{where $\displaystyle\zeta(s):=...
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Scattering amplitudes and the Riemann zeta function

I'm reading Amplitudes and the Riemann Zeta Function, which recently appeared in Physical Review Letters. It's received some publicity, including my own campus' PR operation. From the abstract (...
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Laplace transform of power of zeta function

Let $s$ is the complex variable. I would like to figure out the region of absolutely convergency of the following integral $$ e^{\frac{is}{2}}\int\limits_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\...
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Moments of the Riemann zeta function

Is it possible to get an upper bound better than $\ll_\sigma T^{3/2-\sigma}$ for $$\int_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$
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Upper bound for moment of the Riemann zeta function

Is it possible to get a good upper bound for the integral $$\int_{0}^{T}\zeta ^{3}(\frac{1}{3}+it)dt$$ (unconditionally)?
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Density of fake zeros of Zeta

I am investigating whether or not there exist $\epsilon > 0$ such that $\zeta(s) \neq 0$ on the strip $1-\epsilon < \Re(s) \leq 1$. Suppose not. Then given $\delta > 0$ there exists a zero ...
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Three questions about three functions similar to $\sin,\cos$

In The Basel problem revisited? a question about the function, similar to sinc, $f(x)$ was asked: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \prod_{n=1}^\infty \left ( 1+ \frac{...
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Growth of residues of $1/\zeta(s)$: conjectures?

Let $\rho$ range over the non-trivial zeroes of the Riemann zeta function. Let $$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| = \max_{|\Im \rho|\leq T} \frac{1}...
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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{\...
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Speed of convergence of $\zeta(2k)\to 1$?

From the definition of $\zeta(z):= \sum_{k=1}^\infty \tfrac{1}{k^z}$ for $\mathrm{Re}(z)>1$ it is obvious that $\zeta(2k)\downarrow 1$ as $k \rightarrow \infty$. I am interested in the "true&...
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A paper by Viggo Brun

Does any one know a digital link to Viggo Brun's paper: "Deux transformations elementaires de la fonction Z de Riemann" Revista Ci. Lima 41 (1939), 517-25. ??
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The Basel problem revisited?

In the Basel problem, the $sinc$ function is considered at the Wikipedia page. Let me try to make an alternative function definition: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
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6 votes
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Best explicit bound on $\zeta'(1+it)/\zeta(1+it)$

Assume the Riemann hypothesis. We know that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| \leq 2 \log \log t + O(1)$$ (see, e.g., Thm. 13.13 in Montgomery-Vaughan). What is the best explicit bound ...
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Conditional results on average size of Mertens' function

Let $M(x) = \sum_{n \le x} \mu(n)$ where $\mu$ is the Möbius function. Titchmarsh, in his book on the Riemann zeta function, considers consequences of the hypothesis that $$\int_{1}^{X} \left( \frac{M(...
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A kind of reflection formula for the logarithmic derivative of the zeta function

So I was messing around with Bernoulli numbers and values of $\zeta'$ at integers $-$ and suddenly I came about a non trivial identity which can be written in terms of the logarithmic derivative of ...
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Is there any literature on $\sum_{i=1}^{k} \left[ {k \atop i} \right] \zeta(i+1) $?

As per these questions, I'm trying to evaluate $$\sum_{n=2}^{\infty} \big{(} \zeta(n)^{2}-1 \big{)} = 1+ \sum_{m=2}^{\infty} \frac{H_{-\frac{1}{m}}}{m}. $$ Here, $H_{x}$ is a generalized Harmonic ...
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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 ...
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Is it possible in principle (but not in practice) to recursively factor away the Riemann zeta zeros as they are computed?

Let: $$f_0(x)=\frac{\zeta (x)}{\sin \left(\frac{\pi x}{2}\right)}$$ and let the seed point be: $$s=\sqrt{-1}$$ which is the input into the limit: $$\rho_1=s+\lim\limits_{n \rightarrow \infty}\left(1-\...
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Approximation for the $n$th nontrivial zero of $\zeta(s)$

For all positive integers $n$, let $$t_n = \frac{1}{2\pi} \operatorname{Im} \rho_n,$$ where $\rho_n$ donates the $n$th nontrivial zero of the Riemann zeta function in the upper half plane (listed in ...
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2 votes
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Riemann-Von Mangoldt formula, revised question

This is my last question, building off of Riemann-Von Mangoldt formula and Does $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t)$ have mean value $0$?. I apologize for taking a while to ...
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7 votes
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Does $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t)$ have mean value $0$?

I'm curious about what is known about the distribution of the function $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t) \in (-1,1]$, on a linear or logarithmic scale, where $\operatorname{Arg}$ ...
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8 votes
4 answers
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Riemann-Von Mangoldt formula

Let $$N(T) = \#\{\rho \in \mathbb{C}: \zeta(\rho) = 0,\, \operatorname{Im} \rho \in (0,T]\}$$ denote the number of zeros of $\zeta(s)$, counting multiplicities, with imaginary part lying in the ...
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4 votes
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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 $x$. ...
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On the nearest integer to $\zeta^{(k)}(1-1/B),B \ge 2$

Let $k \ge 1,B \ge 2$ integers and $\zeta^{(k)}(s)$ denote the $k$-th derivative of Riemann zeta function. For real $x$, let $[x]$ denote the nearest integer to $x$. Conjecture 1: For all $n \ge 1,[\...
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On the nearest integer to $\zeta(1-1/B),B \ge 2$

Let $B \ge 2$ be integer and $[x]$ denote the nearest integer to real $x$. For $2 \le B \le 10^5$ computations with mpmath suggest: $$ [\zeta(1-1/B)]=-B+1 \qquad (1)$$ Is (1) true for all $B \ge 2$?
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A expression for the tangent function involving $\zeta(n),n=2,3,\ldots$

A few procrastinal computations motivated by Four infinite series involving Riemann zeta function suggest the identity $$\tan\left(\frac{\kappa-1}{\kappa+1}\frac{\pi}{2}\right)=\frac{1}{\pi}\sum_{n=1}^...
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Consequences of infinitely many double zeros of zeta function of number field

Related to this and this. Let $K$ be the number field with the degree 24 defining polynomial ...
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Bound on $L^2$ norm of $1/\zeta(1+i t)$?

What sort of bounds (explicit of preference) can one give for $$\int_T^{2 T} \frac{dt}{|\zeta(1+i t)|^2} \;\;\;\;\;?$$ Some obvious points: One can give a pointwise bound $\frac{1}{|\zeta(1+ it)|} \...
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On sum over non trivial zeroes of riemann zeta function

I wanted to know if there is an estimate or any closed form on the following partial sum series $$\sum_{n=1}^k \frac{1}{|\alpha_n||(\zeta'(\alpha_n))|}$$ Where "$\alpha_n$" runs over all non-...
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20 votes
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A multiple integral that seems related to the $\zeta$ function at even integers

I came across this integral that seems related to the Riemann zeta function $\zeta(2n)$ evaluated at even integers $2n \in 2\mathbb{Z}$. Letting $n$ be an even integer, define the multiple integral ...
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1 answer
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Integrals involving $1/|\zeta(1+i t)|^2$: closed expressions?

Is there by any chance anything resembling a closed expression for, say, the integral $$I = \frac{1}{2 \pi} \int_{-\infty}^\infty \frac{dt}{|\zeta(1+i t)|^2 t^2} ?$$ It is easy to show (by Plancherel) ...
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6 votes
1 answer
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Asymptotic estimate for an integral involving the squared modulus of the Riemann zeta function

For any fixed $\frac{1}{2} < \sigma < 1$, let $$\int_0^T \frac{|\zeta(\sigma+it)|^2}{\sqrt{1+t^2}} \ dt = O(T^\theta), \qquad T \uparrow \infty. $$ It is clear that $\theta > 0$, since we ...
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2 votes
1 answer
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Proof of the sum of the reciprocal non trivial zeros cubed

Just for fun I was trying to find a formula that calculates the value of the sum of the Riemann zeta non trivial roots raised to a power $n$, $Z(n)$. $$Z(n) = \sum_{\rho} ' \frac{1}{\rho ^n}$$ I ...
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21 votes
2 answers
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What are the consequences of an ineffective proof of the Riemann Hypothesis?

Suppose a proof came out (and was verified by credible peer review) of the following statement: There is a $T_0$ such that for all $t>T_0$, all zeros $\zeta(\beta+it)=0$ have $\beta=1/2.$ where $...
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