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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Regularizing the divergent sum $1^k + 2^k + \cdots$
EDIT:
Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$
I was looking at ...
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PT Symmetry and the Riemann Hypothesis
Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...
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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
$$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| =
\max_{|\Im \rho|\leq T} \frac{1}...
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Witten zeta function v.s. Riemann zeta function
From a talk, we learned that
The symplectic volume of the space $M$ of gauge equivalence classes of flat G connections is given by the “Witten zeta function”:
where we sum over irreducible ...
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Computing Mertens' function in time O(sqrt(x)) - in practice
As far as I know, there is one way currently known to -- in principle -- compute the Mertens function $M(x) = \sum_{n\leq x} \mu(n)$ in time essentially $O\left(x^{1/2}\right)$, namely, a modification ...
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The zeta function and classical mechanics
In this paper, Guilherme França and André LeClair show that $$\gamma_{y}\sim 2 \pi \left(y-11/8\right)/W\left((y-11/8)e^{-1}\right)$$ where $W$ is the Lambert W function, and $\gamma_{y}$ is the ...
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Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$
If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
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New series for $\zeta(5)$ involving second-order harmonic numbers
In 1997 T. Amdeberhan and D. Zeilberger proved that
$$\sum_{k=1}^\infty\frac{(-1)^k(205k^2-160k+32)}{k^5\binom{2k}k^5}=-2\zeta(3).\tag{1}$$
In 2008 J. Guillera obtained that
$$\sum_{k=1}^\infty\frac{(...
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Numerical coincidence for the first Riemann zero
Is there any heuristic explanation for why the imaginary part of the first nontrivial Riemann zero, $\gamma_1\approx 14.134725$, should be so close to $\pi/2$ modulo $2\pi$? Mathematica gives ...
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Bounding $1/\zeta(s)$ given RH
Let $T\geq 0$. Assume RH(T+100), that is, assume that all non-trivial zeros $\rho$ of the Riemann zeta function with $|\Im(\rho)|\leq T+100$ satisfy $\Re(\rho)=1/2$. Can we then give a good upper ...
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Implications of divergence of $1/\zeta(s) $ at 1/2
$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function.
This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$.
Its convergence is unknown if $1/2< s&...
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From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
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Can this infinite sum for the Riemann zeta function be generalised?
I recently derived the following identity (which is probably a rediscovery of something well-known to experts).
$$\sum_{k=1}^\infty{\frac{(-1)^{k+1}k^{4n+1}}{e^{k \pi}-(-1)^k}}=\frac{1}{2}\zeta\left(-...
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On Riesz criteria for Riemann hypothesis:
Marcel Riesz defined a function :
$R(x) = \sum_{n=1}^\infty \frac {(-1)^n x^n} {\zeta(2n)\Gamma(n)}$
The Riemann hypothesis holds if $R(x)= O( x^{1/4 + {\varepsilon}}$)
For any $\varepsilon$
We have ...
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Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?
Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
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Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?
Numerical evidence suggests that the complex zeros of:
$$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$
all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...
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Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$
Note: This question has been brought here from MSE.
I have been working on various sums involving the zeta function (which come up frequently in my research), and it turned out that many of them had ...
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Order of magnitude of extremely abundant numbers and RH
I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion (you can find the scanned paper here) that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \...
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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, ...
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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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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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"Non-critical" zeros of $\zeta$ and the $\zeta$-cycles of Connes and Consani
In the recent preprint of Connes and Consani https://arxiv.org/abs/2106.01715 a new spectral realization of the critical zeros of $\zeta$ (edit: defined as being those on the critical line only, see ...
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Computability assertions for Riemann zeta zeros
While looking for information about the Riemann zeta function, I kept running into the claim that there is an algorithm to decide whether or not a zero of the function is off the half-line. Is this ...
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Is there a connection between the sequence of a finite number of Stieltjes constants and the integer partitions number?
Lehmer 1988 and Keiper 1992 made major progress on evaluating the series:
$$\sigma_r = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^r} + \frac{1}{(1-\rho_k)^r}\right) \quad r \in \mathbb{N}$$
where $\...
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Fully explicit version of Atkinson's formula?
Let $$I(T)=\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt$$
and let $E(T)$ be $I(T)$ minus what turn out to be its main terms:
$$E(T) = I(T)- T \log \frac{T}{2 \pi} - (2 \gamma - 1) T.$...
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Alternative approaches to Zudilin's proof of Apéry's theorem
The article
Wadim Zudilin, Apéry's theorem. Thirty years after, Int. J. Math. Comput. Sci. 4 (2009), no. 1 pp 9–19. (arXiv:math/0202159, with the title An elementary proof of Apéry's theorem)
...
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When does the function $F(x)=\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$ reach $F(x) > 8$?
We know from Ramanujan and Riemann that,
$$\pi(x) = \operatorname{li}(x) -\tfrac12\operatorname{li}(x^{1/2})-\tfrac13\operatorname{li}(x^{1/3})-\tfrac15\operatorname{li}(x^{1/5}) +\dots$$
with prime ...
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How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?
I like to expand on this (unanswered) MSE question.
Take the following, nicely symmetrical, telescoping series for $\zeta(s)$:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum _{n=1}^{\...
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How comes vanishing of the real part of function involving zeta is very well approximated by algebraic curve?
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:
$$
f(s)=\Re \left( \zeta\left(\frac{...
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Are these identities Newton series?
Newton series is the following expansion of a function:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$
Now ...
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$\zeta(x)$ in terms of $\zeta'(x),\zeta'(1-x),\Gamma,\psi$
By differentiating $\xi$ and solving for $\zeta(1-x)$:
$$ \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)-\...
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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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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
$$\int_{1}^{X} \left( \frac{M(...
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A kind of reflection formula for the logarithmic derivative of the zeta function
So I was messing around with Bernoulli numbers and values of $\zeta'$ at integers $-$ and suddenly I came about a non trivial identity which can be written in terms of the logarithmic derivative of ...
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Generalized prime number theorem and Riemann Hypothesis for non-number math objects
My question is about some math objects (matrices, polynomials) and operators that satisfy a number of properties which can lead to a theory similar to PNT, RH, Dirichlet functions, abscissa of ...
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Value of $\zeta(3/2)$?
Is anything known about the value of $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}?$$ It is a classical result that $\displaystyle \zeta(2)= \frac{\pi^2}{6}$ and $\zeta(3)$ has been shown to be ...
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Is there a conjectured uniform Lindelof hypothesis for Hurwitz zeta functions
Consider $\zeta(s, a) = \sum_{n=1}^{\infty} (n+a)^{-s}$ (alternatively, consider its functional equation Dirichlet series $\sum_{n=1}e(a n) n^{-s}$). What is the expected growth-rate of $\zeta(1/2 + ...
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Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?
Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of:
$$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$
are on the ...
6
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An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function
I think, here, I found
$$
P_x(s)=\sum_{p < x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n}
\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}}
\left[
{\rm li}(t^{\frac zn-s})
\right]^{x}_2
\tag{7}
$...
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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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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 (...
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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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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 ) = \...
5
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Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?
In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
5
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On the asymptotics of some sum involving the Mertens function
Let $a_n$ be a sequence of nonnegative real numbers such that $\sum_{n\leq x} a_n \gg \frac{\sqrt x}{\log x}$ for large enough $x$. Denote by $\mu$ the Mobius function, and let $M(N)=\sum_{n\leq N} \...
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Are the ordinates of the non-trivial zeros of $\zeta(s)$ uniformly distributed around the mid points of Gram point intervals they can be mapped to?
Let $\rho_n$ be the $n$-th non-trivial zero of $\zeta(s)$ and $z_n = \Im(\rho_n)$ with $z_n > 0$ and $z_{n+1} \ge z_n$.
A well known method to establish that all $\rho$s reside on the critical line ...
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Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?
This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO.
For every ...
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Remainder term in an integral linked to the Riemann zeta function
Sorry if this is not research level, but the following problem occurs in my own research: it is trivial to show that for $k\ge2$ integral we have
$\zeta(k)=(1/(k-1)!)\int_0^\infty t^{k-1}/(e^t-1)\,dt$ ...
5
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Dirichlet eta function and Stirling Permutations
The Stirling permutations of order $k$ is a permutation of the multiset $1, 1, 2, 2, ..., k, k$. The Dirichlet $\eta$-function is a function closely related to the Riemann $\zeta$-function.
According ...