Questions tagged [riemann-zeta]
Questions on the famed $\zeta(s)$ function of Riemann, and its properties.
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How to evaluate $\sum_{n=2}^{\infty}\left(\frac{1}{\zeta (n)}-1\right)$?
It is well known and easy to evaluate, that
$\sum_{n=2}^{\infty}\left(\zeta (n)-1\right)=1$. I am trying to evaluate $$\sum_{n=2}^{\infty}\left(\frac{1}{\zeta (n)}-1\right)$$
The classical way gives ...
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Empirical study of zeroes of RIemann Zeta Function [closed]
Has there been any study done on how the imaginary part of the zeroes are distributed ?
Is there a closed form for a first zero ? is there a generating function ?
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$\zeta(z)$ Identities and a Conjecture
Brief Introduction
I found something quite interesting and would like to share it here.
The surface area and volume of a unit radius $n$-sphere can be expressed in terms of the gamma function $\Gamma(...
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Evaluate $\zeta(0)$
I am using the equation
$$\zeta(s) = \frac{\pi^{s/2}}{2 \Gamma(s/2)} \int_1^\infty (\theta(t)-1)\left(t^{s/2} + t^{(1-s)/2}\right) \frac{dt}{t} + \frac{\pi^{s/2}}{(s-1)\Gamma(s/2)} - \frac{\pi^{s/2}}{...
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how to calculate the arg(ζ(0.5+iT)) where ζ is the zeta function [closed]
The problem is that if σ + iτ is a zero of ζ"zeta function" such that 1 − α < σ < 1 and 0 < τ < T (ζ has a zero outside the rectangle D(α,T) with vertices A = (1−α,iT),B = (α,iT),...
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$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i}\frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)ds$ as a finite sum of $\Lambda(n)$
[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 13, question 7]
Express
$$\frac 1{2\pi i}\int_{2-\infty i}^{2+\infty i} \frac{x^s}{s^2}\left(-\frac{\zeta^{\prime}(s)}{\zeta(s)}\right)...
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Why doesn't the Riemann Zeta Function have zeroes at positive even integers?
According to the Riemann Functional Equation (source: https://en.wikipedia.org/wiki/Riemann_zeta_function) the Zeta Function is equal to itself multiplied by a bunch of stuff, including $$sin(πs/2)$$ ...
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Dirichlet convolution inverse of Euler's totient function
Let g(n) be the convolution inverse of Euler's totient function $\varphi(n)$. Let $n=p_1^{a_1}...p_t^{a_t}$, where $p_j$ are the distinct prime divisors of $n$. Find a formula for $g(n)$ and prove ...
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Is $|\pi(x) - Li(x)| < \frac{\sqrt{\ln(x) x}}{8 \pi}$ for $x>q$. for some fixed $q$ still potentially consistant with RH?
Let $\pi(x)$ be the prime counting function.
From the question
If $\pi(x) = Li(x) + O(\ln^3(x) \sqrt x) $ is true, what does that say about the Riemann zeta zero's?
came the related question :
Is
$...
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Proving that $\int_{0}^{1}\left(\zeta(t)+\frac{1}{1-t}\right)dt=\sum_{n=0}^{\infty}\frac{\gamma_{n}}{(n+1)!}$ [closed]
It seems that the above identity is true. Can this be proven? Or are there references treating sums like the right hand side?
The above constants, $\gamma_{n}$, are the Stieltjes constants.
Thanks.
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If $\pi(x) = Li(x) + O(\ln^3(x) \sqrt x) $ is true, what does that say about the Riemann zeta zeros?
Let $\pi(x)$ be the prime counting function.
The statement $\pi(x) = Li(x) + O(\ln^3(x) \sqrt x) $ is in "essence" weaker than what we can conclude from the RH if the RH is true.
I assume it ...
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Why do fractal-like patterns appear in this sequence?
I came across this sequence called Digital River, where the next number in the sequence is defined as the sum of the digits of the previous number, plus, the previous number itself.
It caught my ...
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For arithmetical periodic function $f$, if $\sum_{r=1}^k f(r)=0$, then $S=\sum_{n=1}^\infty \frac{f(n)}{n^{s}}$ converges
[Introduction to Analytic Number Theory - Tom M. Apostol, chapter 12, question 1(b)]
Let $f(n)$ be an arithmetical function which is periodic mod $k$. If
$$\sum_{r=1}^k f(r)=0$$
then prove that the ...
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"Mollifier" of the Dirichlet L-function
I was studying some zero-density results for $\zeta(s)$, mostly from Titchmarsh's book "The Theory of the Riemann zeta function", Chapter 9. In one place, as per the literature, a mollifier ...
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Proofs involving manipulation of divergent series
Is this proof valid even though the harmonic series it is based on is divergent?
Prove: $$\sum_{n=2}^\infty (\zeta(n)-1)= 1$$
Where $\zeta$ as in Riemann's Zeta function is summed over all natural ...
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$ 0 = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)} \implies Re(s) \leq \frac{1}{2}$?
Define $f(s)$ as
$$ f(s) = 1 + \sum_{n=2}^{\infty} \frac{n^{-s}}{\ln(n)}$$
where we take the upper complex plane as everywhere analytic.
Notice this is an antiderivative of the Riemann Zeta function, ...
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Question concerning an assertion regarding the modulus of the Riemann Zeta function (follow up)
Update Nov 25th-- I made a few small changes, especially in my assertion of "Lemma 1". I have yet to find the final series of telescoping and collapsing summands described at the end below.
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Series involving derivative of Riemann Zeta function: $\displaystyle \sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$
1. Question
Could anyone recommend a useful method for approaching the following series?
$$\sum_{k=2}^{\infty}\zeta'(k)x^{k-1}$$
Where $\zeta(z)$ is the Riemann Zeta function.
I've seen that there are ...
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Convergence of $\sum_{\rho} |\Gamma(\rho)|$
In his proof of the prime number theorem Littlewood claims that the sum $\sum_{\rho} |\Gamma(\rho)|$ converges, where ${\rho}$ ranges over the non-trivial zeros of $\zeta$ counting multiplicity.
I ...
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Interchanging sum and integral (analytic continuation of zeta)
While learning about analytic continuation of $\zeta(s)$, I have come across the following formula
$$\pi^{-s/2} \Gamma(s/2)\zeta(s) = \sum_{n=1}^{\infty} \int_{0}^{\infty} e^{-n^2 \pi x} x^{s/2 -1} dx$...
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Analytic functions such that $\sum_{n=2}^{\infty} f(n) \left( \zeta(n)-1 \right)$ can be evaluated, but $\sum_{n=2}^{\infty} f(n)$ can't
Background
The general context for this question is the topic of rational zeta series. What I've found so far, is that it usually the case that sums of the form $$\zeta_{f} := \sum_{n=2}^{\infty} f(n) ...
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What's so special about the Riemann Zeta zeros being on the $x=\frac{1}{2}$ line?
There are lots of explanations (videos, blogs) on the Riemann Zeta function online.
They all describe analytic continuation beyond the $x=1$ line and then the fact that there are trivial zeros and ...
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Question concerning an assertion regarding the modulus of the Riemann Zeta function
Posting this question here as the community has addressed the Riemann Zeta function with questions such as, i.e. with Simpler zeta zeros and Zeta function zeros and analytic continuation.
Define the ...
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Functional equation for $\theta$ function using functional equation of $\zeta(s)$
Let $$\theta(t) = \sum_{n \in \mathbb Z}e^{-n^2 \pi t}.$$ We can derive a functional equation for $\theta$ using Poisson summation formula: $$\theta(1/t) = \sqrt t \theta(t). $$
Riemann uses the above ...
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Asymptotic Behavior of Riemann Zeta Function
I am currently reading an article that claims that the value $$\inf \{ a \in \mathbb{R}, |\zeta(\sigma + it)| = O(|t|^{a}) \}$$ is zero if $\sigma > 1$ and is $\frac{1}{2}-\sigma$ if $\sigma < 0$...
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Riemann zeta conjugate extension to real plane < 1
I found this function:
$$
\zeta(\alpha+i\beta)*\zeta(\alpha-i\beta)=\prod \frac{1}{1-\frac{2cos(\beta ln{p})}{p^\alpha} + \frac{1}{p^{2\alpha}}}=\Re(\zeta(\alpha+i\beta))^2 +\Im(\zeta(\alpha+i\beta))^...
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Applying Euler's trick for the Basel problem to cos leads to a wrong result?
I'm trying to do what Euler did with the $\sin(x)$ for the Basel problem, but for $\cos(x)$. This seems to be leading to an incorrect result.
First, the Taylor series of $\cos(x)$ about $x=0$:
$$\cos(...
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Using Riemann's methods to derive the floor function of x from the zeta function
I thought it would be interesting to try and use Riemann's methods to derive (rather than the quantity of primes less than $x$) the quantity of natural numbers less than $x$. This function is, of ...
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Relationships between Riemann Xi and Jacobi Theta functions
I asked ChatGPT about the relationship between the Riemann Xi function and the Jacobi theta functions. ChatGPT provided the following relation:
$\xi\left(\frac{1}{2} + it\right) = \frac{1}{2} \left( \...
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Explore the relationship between $\sum\limits_{n = 1}^{2x} \frac{1}{{n^s}^x}$ and $\sum\limits_{n = 1}^{2x-1} \frac{(-1)^{n-1}}{{n^s}^x}$ [closed]
I am trying to find an algorithm with time complexity $O(1)$ for a boring problem code-named P-2000 problem. The answer to this boring question is a boring large number of $601$ digits. The DP ...
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What does the zeta squared function converge to
Define a series: $f(n) = \frac{1}{n^2}$ where $n$ is a natural number. This series converges to $0$.
Now, consider the series $S(n) = \sum\limits_{i=0}^n f(i)$. This is the Reimann Zeta function ...
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Using Ramanujan-type series for $1/\pi^m$ to find formulas for $\zeta(2),\, \zeta(3),\, \zeta(4)$?
As described by Guillera in "Ramanujan Series with a Shift", one nice thing about Ramanujan-type $1/\pi^m$ formulas is by "shifting" them, they can yield a second value which may ...
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Integrability of the Jacobi Theta Function
Let $$\psi(x) = \sum_{n = 1}^{\infty} e^{-n^{2} \pi x}$$ be a theta function. Can it be shown that that
$$\int_{0}^{\infty} \psi(x) \cdot dx < \infty$$ without invoking Fubini-Tonelli’s Theorem ...
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Evaluate $\zeta(2)$ by limit
Ever since I came over this method of finding a limit:
Example: Find the limit of $f(n)=\cfrac{1+2+3+...+n}{n^2}$ as $n\rightarrow \infty$, which turns out to be as follows
Since the limit exists, let ...
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The ultimate polylogarithm ladder
As you can see, here I performed a derivation of a quite simple formula, not much differing from the standard integral representation of the Polylogarithm. Seeking to make it fancier, I arrived at ...
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Does this series representation for $\zeta(s)$ converge in the critical strip?
The Wiki page of the Riemann Zeta function contains a series representation for $\zeta(s)$ involving the rising factorial. This representation could also be written as:
$$\zeta(s)= \frac{s}{s-1}-\...
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Series involving $\sum_{m,n}\frac{1}{(m^2+n^2)^{2k}}$
The series
$$\sum_{k=1}^\infty \zeta (2k)z^{2k}$$
(where $\zeta$ is the zeta function) has a non-zero radius of convergence (in $z$). This led me to the following question:
Let
$$C_k=\sum_{m=1}^\infty\...
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Are there any real analytic functions such that $\sum_{n=1}^{\infty} \frac{1}{f(n)} = \sum_{k=2}^{\infty} f(k) \left( \zeta(k) - 1 \right) $?
I wonder whether there are any real functions that are analytic on $\mathbb{R}_{\geq 1}$. that satisfy the equation $$\sum_{n=1}^{\infty} \frac{1}{f(n)} = \sum_{k=2}^{\infty} f(k) \left( \zeta(k) - 1 \...
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Simplifying a laurent series involving inverse squares
If I write
$$
H(x) = \frac{1}{x^2} + \frac{1}{(x+1)^2} + \ldots + \frac{1}{(x+k)^2} + \ldots
$$
I can see that finding $H(1)$ amounts to the Basel problem, which makes me a bit pessimistic, but...
Is ...
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Is this inequality related to $|\log \zeta (s)|$ wrong?
This is a doubt in the Littlewood's first estimate(in which it is assumed that Riemann hypothesis is true) given in this book, page 433.
Let $s = \sigma + it$ and $\delta $ such that $\frac12 + \...
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Proof explanation that the Riemann hypothesis implies $L(x) = O(x^{\frac12 + \varepsilon})$, where $L(x) = \sum_{n\leq x}\lambda(n)$
$\lambda(n)$ is the Liouville function. According to the first proof in the paper "The distribution of weighted sums of the Liouville function and Pólya’s conjecture", the Riemann hypothesis ...
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Logarithmic Integral evaluation
I was watching this video explaining the Riemann hypothesis, and there is the use of the Li(x), Logarithmic integral function: https://youtu.be/GEcHgadgOn4
It seems that this integral cannot be ...
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Why does Riemann's prime counting $\Pi_0(x)$ function jump by 1/2 when x is exactly a prime?
The Riemann prime counting function in question is the following:
\[\Pi_0(x)=\frac{1}{2}\sum_{p^k<x}\frac{1}{k}+\frac{1}{2}\sum_{p^k\leq x}\frac{1}{k}\]
So that, for any $x$ where $x$ is exactly a ...
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How do we increase the region of convergence for the Riemann Zeta function (using Dirichlet Series form)?
The Riemann Zeta Function can be defined as: $\zeta(s)=\sum \frac 1 {n^s}$ for $s>1$. The series converges for $s>1$. wiki (https://en.wikipedia.org/wiki/Riemann_zeta_function) mentions that:
An ...
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Approximating the logarithms of primes elegantly
What's the "most efficient" way to encode that $\ln(2):\ln(3):\ln(5):\ln(7) \approx 171:271:397:480$ using $3$ approximations?
For example:
$((\frac{3^3}{5^2})^3)^3 \approx 2$ uses $3+2+3+3 ...
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Implication of the zero free region of the $\zeta$-function to the von Mangodlt form of the prime number theorem
I am currently trying to understand a proof of the prime number theorem of Terence Tao. I think I understand nearly everything but the proof of the implication of the theorem, that $\zeta(s)$ has no ...
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Derivatives of Euler products
Suppose I have an Euler product absolutely convergent for $\sigma >1$ $$\mathcal Z_\mathbf z(s)=\prod _p\left (1-\frac {1}{2p^{s}}\left (\frac {1}{p^z}+\frac {1}{p^{z'}}\right )\right );$$ note $$\...
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Why does reordering the terms of an infinite series make it convergent
Why does rearranging terms in an infinite series make two series convergent and divergent for the same values. Shouldn't the sum be the same. After all we are not adding new terms. Can someone explain ...
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For what values of $c$ is $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}=0$ when $x=\exp \left(-\frac{\rho _1}{c}\right)$?
The alternating Dirichlet series, the Dirichlet eta function, can be written in the form: $\sum _{k=1}^{\infty } (-1)^{k+1} x^{c \log (k)}$
For what values of $c$ is $$\sum _{k=1}^{\infty } (-1)^{k+1}...
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Proof of the Functional Equation of Riemann Zeta Function
I am reading the book "Riemann's Zeta Function" by H. M. Edwards. I had a confusion in Section 1.6, Page 13 at a proof of the functional equation of Riemann Zeta Function.
It has been ...
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