Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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Any complex analysis book with programming assignment and exercises?

All: I had studied complex analysis long time ago. Now, I would like to review some material, particularly about Analytic function, Riemann zeta and Analytic function. I have been a software ...
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79 views

Limit of differences of truncated series and integrals give Euler-gamma, zeta and logs. Why?

In the MSE-question in a comment to an naswer Michael Hardy brought up the following well known limit- expression for the Euler-gamma $$ \lim_{n \to \infty} \left(\sum_{k=1}^n \frac 1k\right) - ...
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A cotangent series related to the zeta function

$$\sin x = x\prod_{n=1}^\infty \left[1-\frac{x^2}{n^2\pi^2}\right]$$ If you apply $\log$ to both sides and derivate: $$\cot x = \frac{1}{x} - \sum_{n=1}^\infty \left[\frac{2x}{n^2\pi^2} ...
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Difference between Eulers product and Zeta Function at a finite values

So a very important formula proven by Euler is that is equal to of course these formulas give you the same value when they reach infinity, but my question is that say s=1. What would be the ...
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89 views

Why there's no articles about the eta function convergence?

I've been searching about a proof that the eta function converges for $\mbox{Re}(z)>0$ but the ONLY page I've found that claims to prove it was in this question: ...
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1answer
60 views

Riemann Zeta Function On Line Re(s)=1

I am having trouble thinking about this. Since the Riemann Zeta Function is analytic everywhere except at $s=1$, it follows that it is continuous on the real line $Re(s)=1$ except at $s=1$. Now, the ...
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Values of derivatives of Jacobi theta function

The Jacobi theta function is defined as: $$\theta(x)=\sum_{n=-\infty}^{\infty}\exp(-\pi n^2 x)\text{ }, x>0$$ On wikipedia.org, we can find close-form expressions for the values of ...
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Are there any Riemann zeta like functions that may have nontrivial zeros on the critical line but only involves integers up to $N$?

Question: Do there exist any Riemann zeta $\zeta(s)$ like functions $f_N(s)$ that may have all nontrivial zeros (verified via numerical calculation) on the critical line but only involve integers up ...
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Riemann Zeta Function Analytic Continuation

I am struggling to understand how the analytic continuation of the Riemann Zeta function is derived to extend it to all complex values $z$ not equal to $1$, starting with the series which converges ...
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What is the sign of the generalized Stieltjes constants $\gamma_{k}(a)$?

Recall that the Stieltjes constants $\gamma_{k}$ appear as the coefficients in the regular part of the Laurent expansion of the Riemann zeta function about $s = 1$: $$ \begin{align} \zeta(s) = ...
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174 views

Connection between integral expression and the factorial of infinity

Does the fact that $$\int_{-\infty}^{\infty}\exp\left(-\frac{1}{2}x^2\right)\mathrm{d}x=\sqrt{2\pi}$$ Have something to do with the fact that the regularized factorial of infinity is also ...
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91 views

Dirichlet series and Riemann zeta function

Im trying to show, for $\Re(s)>1$, that $\displaystyle\sum_{n=0}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)}$, where $d(n)= |\{k \mid k|n \}|$, number of positive integers that ...
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213 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
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104 views

What does this complex contour integral represent?

How would one evaluate the following complex contour integral in "Integral and Series Representations of Riemann’s Zeta function, Dirichelet’s Eta Function and a Medley of Related Results." The ...
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221 views

A series $\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\frac{(i-1)! (j-1)!}{(i+j)!}H_{i+j}$ and $\zeta(3)$

We have $$ \sum_{j=1}^{\infty}\sum_{i=1}^{\infty} \displaystyle \frac{(i-1)! (j-1)!}{(i+j)!} H_{i+j} =\displaystyle 3 \: \zeta(3) $$ where $\displaystyle H_{n}:=\sum_{1}^{n} \frac{1}{k}$ are ...
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From the series $\sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)$ to $\zeta(\frac{1}{2}+it)$

Here is a pretty series $$ \displaystyle \sum_{n=1}^{+ \infty} \left(H_{n}-\ln n-\gamma -\frac{1}{2n}\right)=\frac{1}{2} \left(1-\ln (2\pi)+\gamma\right) \quad (*) $$ where $H_{n}:=\sum_{1}^{n} ...
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108 views

Convergence of Euler-transformed zeta series

I am trying to prove that the expression $$(1-2^{1-s})\zeta(s)=\sum_{n=0}^\infty\sum_{m=0}^n\frac{(-1)^m}{2^{n+1}}{n\choose m}(m+1)^{-s}$$ converges to an analytic continuation of the alternating ...
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1answer
113 views

Playing fast and loose with divergent series [closed]

I have been playing around recently with the regularization of infinite divergent sums and products, e.g. $$1+1+1+1+1+\ldots=\zeta(0)=-\frac{1}{2}$$ $$1+2+3+4+5+\ldots=\zeta(-1)=-\frac{1}{12}$$ ...
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A modular form related to Jacobi theta function

The Riemann $\Xi(z)$ function is defined as $$\Xi(z)=2\int_1^\infty A(x)x^{-1/4}\cos\left((1/2)z\ln x\right)dx$$ where $A(x)$ is given in terms of Jacobi theta function ...
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Is there a nice function representation of $\sum_{n=1}^\infty \zeta(2n+1)x^{2n+1}$

$$\sum_{n=1}^\infty \zeta(2n)x^{2n} = -\frac{\pi x}{2}\cot(\pi x) $$ Does $$\sum_{n=1}^\infty \zeta(2n+1)x^{2n+1}$$ have a nice function representation as well? From its graph, it looks like a ...
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106 views

Probability that two random integers have only one prime factor in common

The probability that two integers picked at random are relatively prime is known to be $1/\zeta{(2)}=6/\pi^2\approx0.607927...$. Generalizing, the probability that $n$ random integers have $\gcd=1$ is ...
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Riemann functional equation question?

I was looking through the derivation of the Riemann functional equation, and I understand how to obtain the result $$ \pi^{-\frac s2} \Gamma (\frac s2) \zeta(s) = \pi^{-\frac{1-s}{2}} ...
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37 views

zeros of Incomplet Gamma function

for which values of complex variable $z$ let us getting the zeros of incomplet gamma function ($\Gamma(0.5,z)$) ? I would be interest for any replies or any comments
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50 views

Logistic function approximation of the real valued Riemann $\zeta(x)$ function

Given the function: $$f(x)=\dfrac{a}{1-b\exp(-cx)}+d$$ where: $a = 0.7071$, $b = 2.21$, $c = 0.7672$, $d = 0.2942$, I found the following inequality: $$|\zeta(x) - f(x)|\lt \epsilon$$ for ...
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131 views

Analytic Continuation of Zeta Function using Bernoulli Numbers

In my complex analysis textbook by Stein and Shakarchi, as an exercise, I am supposed to extend $\zeta(s)$ to the entire complex plane using Bernoulli numbers, but I am stuck. I can prove that $$ ...
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Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
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An interesting identity involving powers of $\pi$ and alternating zeta series

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
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Is this the chord G Major I am hearing as base tones from interference of zeta zeros times eigenvalues of the von Mangoldt function matrix?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
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Why doesn't the functional equation imply that $\zeta(s)=0$ for positive even integers?

The Riemann Zeta Function satisfies the functional equation $\zeta(s)=2^s\pi^{s-1}\sin\left(\dfrac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$. But when $s$ is a positive even number, $\sin\left(\dfrac{\pi ...
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2answers
59 views

Values of Incomplete gamma function

Look this claim : Does $\Gamma(0.5,-x^2)= i\alpha$, for $x$ large real number? i=unity imaginary part $\alpha$ is real number I would like someone to prove me this if it's a true claim
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1answer
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An inequality of riemann zeta function

I need show this inequality Let $\sigma >1$, show that $$\zeta(\sigma)\geq \frac{\sigma^2}{(\sigma-1)(2\sigma-1)}$$ Any help is appreciated Thanks!
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68 views

Finding $\zeta(4)$ by Taylor series

Is it possible to solve Zeta(4) function using something similar to the solution for zeta(2) as seen in this video? https://www.youtube.com/watch?v=mTPKyC3Udns
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127 views

Category theoretic approaches to Riemann Hypothesis?

I was wondering if there has been any category theoretic advancements in the study of the Riemann Hypothesis and the theory surrounding it? This question is meant in the same vein as these ...
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Question about the convergence of an infinite series in all of $\mathbb{C}$.

Does the following infinite series: $$G(s):=\displaystyle \sum^{\infty}_{n=1} \frac{(-1)^{n+1}}{n^s + \frac{1}{n^s}}$$ converge for all $s \in \mathbb{C}$ (especially in the critical strip; since ...
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Are there any new research results on approximating Riemann $\Xi(z)$ by a Fourier transformation

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ The functional equation for $\zeta(s)$ is equivalent ...
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Riemann Zeta function, quaternions and physics

Disclaimer: This question is rather vague, and thus probably not suitable for Mathoverflow, so I prefer to ask it here. I'm sorry if it doesn't meet the standards of this site. Several years ago, I ...
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Plotting the pair correlation function for the zeta zeros /GUE

I am making a shameless request for instructions on how to plot this: from this page. I can see from here that normalizing the zeros is given by ...
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Riemann Zeta function Analytic continuation integral

Following Riemann paper about analytic continuation of Zeta Function: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf I can't understand the contour integral step: "If one now ...
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Generating function of the squared Riemann zeta function

It's a well known fact that $$\sum_{k=2}^{\infty} \zeta(k) x^k=-x \psi(1-x)-x\gamma \space (|x|<1) $$ but I didn't meet yet a version for squared Riemann zeta function $$\sum_{k=2}^{\infty} ...
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A question about theorem 2 in de Bruijn's 1950 paper “The roots of trigonometric integrals”

Theorem 2 of de Bruijn's paper titled "The roots of trigonometric integrals" (Duke Math. J., 17 (1950)) is given by: What does it mean by "the function $q(x)$ be regular in the sector...? Does it ...
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Question about Riemann's alternating zeta function

Let $\xi$ be Riemann's alternating zeta function. We set $g(\alpha,t)=|\xi(\alpha,t)|^2$ for $\alpha \in (0,1)$ and $t \in R$. Does there exist $t_0>0$ such that for $t>t_0$ the inequality ...
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1answer
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Analytic continuation of Zeta type function

Can one analytically continue the function (Not equal to the Zeta function) $$Z(s)=\prod_{p}\frac{1}{1+p^{-s}}=\sum_{k=1}^{\infty}\frac{(-1)^{\Omega(k)}}{k^s}$$ Where $\Omega(k)$ is the number of ...
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184 views

An integral that might be related to the modified Bessel function of second kind

It is known that the modified Bessel Function $K_z(a)$ ($a>0$)can be expressed as a Fourier transform $$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$ Can ...
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2answers
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Zeta Function $\zeta(1\pm1/n)$ and Euler's constant.

How do I show that $$\lim_{n\to\infty}{\zeta(1+1/n)+\zeta(1-1/n)}=2\gamma$$ and $$\lim_{n\to\infty}{\zeta(1+1/n)-\zeta(1-1/n)}=\infty,$$ where $\gamma$ is the Euler's constant?
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Simpler zeta zeros

Is it true that $$\lim_{y\rightarrow\infty}\dfrac{\sum_{n=1}^{y}n^{-1/2-iy}}{\zeta(1/2+iy)}=1$$ ? Below is a plot of $$\sum_{n=1}^{y}\dfrac{1}{n^{s}}\text{for }s=\dfrac{1}{2}+iy$$ set against its ...
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How to show $\sum_{n=1}^{\infty}\frac{H_{n}}{n^{2}}=2\zeta (3)$? [duplicate]

How to show this equation is true. $$\sum_{n=1}^{\infty}\frac{H_{n}}{n^{2}}=2\zeta (3)$$ where $H_{n}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$
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Why does the imag. part of the graph of $\zeta(n^{ix})$ resemble the tangent function?

If you input $\zeta(n^{ix})$ into the Wolfram Alpha search bar, in the plot, you get an infinitely repeating sinusoidal curve, which resembles the real part, and you get an infinitely repeating ...
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Prime Zeta Function

Does $$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$ imply $$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$ where $H_n$ is ...
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25 views

Explicit formulas for Fourier coefficients from its Tayor expansion

In my research, I need to determine unique coefficients $a_k$ in terms $b_k$: $$\sum_{k=0}^n a_k \cos\left(\frac{k}{n+1}t\right)+O\left(t^{2n+1}\right)=\sum_{k=0}^n b_k t^{2k}.$$ This problem showed ...