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

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A Contradiction of Riemann Zeta Residues

We can show (1+2+3...+n)^2 = 1^3 + 2^3 + ... +n^3, which holds for any finite n, shouldn't this imply Z(-1)^2 = Z(-3)? However, this does not hold if we look at the residues of the zeta function ...
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567 views

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: ...
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1answer
32 views

Riemann hypothesis: An query about the primes [on hold]

How can we describe the Riemann hypothesis easily? What is the connection between the Riemann hypothesis and prime numbers?
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25 views

Plot zeros of partial sum of zeta Riemann with Maple

I want to plot the zeros of the partial sum of the Riemann zeta function with Maple. Some hint?? Thanks!!!
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84 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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1answer
24 views

zero distribution of the Fourier kernel $\Phi(u)$ for Riemann $\Xi(z)$ function

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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How prime numbers are related to special functions?

We know that the Riemann zeta function is defined as $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},$$ for all $\Re(s)>1$. Because of Euler product formula we also know that $$\zeta(s) = ...
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1answer
60 views

the definition of Riemann zeta function

I've read that the Riemann zeta function for $0<s<1$ is defined : $$\lim_{x\rightarrow\infty} \left(\sum_{n \leq x}\frac{1}{n^s}- \frac{x^{1-s}}{1-s}\right)$$ I don't know how to prove that ...
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What do you think of my proof?

I wrote this proof when I was still in high school (I just graduated). Never had the chance to have it checked by anyone. Therefore, I posted it here! :) Please edit if necessary! Our goal is to ...
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1answer
48 views

Zeros of Zeta function and exact roots

Are there exact roots to any of the Zeta zeroes? For example the first one 1/2 +14.134725I, is there a nice looking polynomial that has an exact solution? I would assume if there is an exact value, ...
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500 views

Usage of Complex Numbers in the Riemann Hypothesis.

I don't have a very good understanding of the Riemann Hypothesis, however that being said, could someone explain to me why complex numbers are used, instead of just using real numbers? Everything I've ...
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Mandelbrot set and riemann hypothesis

Has anyone tried to make a connection between the Mandelbrot set and the non-trivial zeros the zeta function? Looking at the Mandelbrot set, it appears that all points are to the left of the line 0.5 ...
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24 views

Singularities of zeta function

I have to prove (if $\gamma \ne 0$) that there is a analytic continuation for $\Re s >0$ of the function $$f(s)=\frac{\zeta (s)^2 \zeta(s-i\gamma )\zeta(s+i\gamma ) }{\zeta(2s)} $$ and that this ...
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1answer
253 views

Analytic continuation of the Riemann zeta function using contour integration

To find the analytic continuation of the Riemann zeta function using contour integration one can integrate $\displaystyle f(z) = \frac{z^{s-1}}{e^{-z}-1}$ around a contour that consists of rays just ...
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Riemann Zeta Function equation

So I'm a amateur mathematician and I was working with the Riemann Zeta Function and I was able to proof this identity. So I'm just wondering if this has already been proven before. For S>0 So I ...
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1answer
265 views

is it possible to get the Riemann zeros

since we know that the number of Riemann zeros on the interval $ (0,E) $ is given by $ N(E) = \frac{1}{\pi}\operatorname{Arg}\xi(1/2+iE) $ is then possible to get the inverse function $ N(E)^{-1}$ ...
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An Inequality Invollving The Riemann Zeta Function

I'm having trouble proving the following inequality for $2<r<3$: $$(1+2^{-r})\frac{(3^r+1)^2}{3^{2r}+1}>\frac{\zeta(r)}{\zeta(2r)}.$$ I can easily plot the graph, and the inequality clearly ...
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1answer
46 views

alternating sum of zeta functions minus one is one half

During my work on a different infinite series I happened to prove that $\displaystyle\sum_{k=2}^{\infty}(-1)^k (\zeta(k)-1)=\frac{1}{2}$ where $ ...
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1answer
52 views

Riemann Zeta of 1/2 $\zeta(\frac{1}{2})$

This may be a silly question, but I need to figure out how to evaluate the value of $\zeta(\frac{1}{2})$. In wikipedia, it says: $\zeta(1/2) \approx -1.4603545$. I am interested to know how this value ...
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3answers
144 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
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2answers
79 views

Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?

Numerical results showed that the series $$Q=\sum_{n=1}^{\infty} (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\epsilon>0\tag{1}$$ with $\epsilon=10^{-6}$ converged to $-0.53259554096828...$ We can ...
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riemann zeta zeros some predictable, some not

Presuming the Riemann Hypothesis, the non-trivial zeros of the zeta function occur when both $\Re\{\zeta (s)\}$ and $\Im\{\zeta(s)\}=0$, where $s=\frac{1}{2}+it$. and since ...
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Question about Riemann zeta function + my proof

First let me say that I am 16 years old so I am not very professional in math. English is also a second language so I apologize for any mistakes. Now i have been reading about the Riemann zeta ...
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Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{\sqrt{n}}$ converge?

Numerical results for $m=1$ to $2000$ showed that the series $$Q(m)=\sum_{n=1}^m (-1)^n \frac{\cos(\ln(n))}{\sqrt{n}}$$ converged to $-0.63986...$ Does the series $$\sum_{n=1}^{\infty} (-1)^n ...
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1answer
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Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. Here is my list: The Riemann Hypothesis: A Resource ...
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1answer
37 views

A question about the convergence of partial products of zeta of one.

Recently I've been toying around with the Totient function and the Prime Number Theorem and came up with the odd result that the following limit $$\lim_{n\to\infty}\frac{\pi(n)m_n}{\phi(m_n)n}$$ ...
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2answers
55 views

Asymptotics for zeta zeros?

What are the best known asymptotics for the nth zeta zero (imaginary part)? Is there anything similar to $p_n\sim n\log n$, ie where $\rho$ is in form $\sigma+it$, $t_n\sim\dots?$
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2answers
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Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$

Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 ...
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how to prove that an entire function is positive on the real axis

The error function $\mathrm{erf}(x)$ is defined as: $$\mathrm{erf}(x):=\frac {2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}dt\tag{1}$$ Let us define the following 3 functions: ...
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Analytic Continuation of the zeta function

Is the analytic continuation of the Riemann zeta function to the upper half plane unique? I don't know much complex analysis, so I can't see why that is the case.
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3answers
517 views

A series involves harmonic number

How do we get a closed form for $$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
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1answer
43 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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Zeta zero sum & reciprocals of prime powers

Below is a plot of $$\dfrac{1}{x}\sum_{n=1}^{75}2\Re\left(\operatorname{Ei}\left(\rho_n\log\left(x\right)\right)\right)$$ where $\rho_n$ is the $n$th zeta zero, with grid lines at primes and prime ...
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Question about the zeros of $\zeta_{H}(s,a) \pm \zeta_{H}(s,1-a)$.

Assume $\zeta_{H}(s,a)$ is the Hurwitz Zeta function. Note that for $a=\frac13,\frac14,\frac16$ the zeros of: $$\zeta_{H}(s,a) + \zeta_{H}(s,1-a)$$ are the same as the non-trivial zeros $\rho$ of ...
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1answer
149 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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80 views

How would you show that the Riemann Zeta function, $\zeta(s) < 0$ for $s \in (0,1)$?

How would you show that the Riemann Zeta function, $\zeta(s) < 0$ for $s \in (0,1)$? So far I have that along the critical strip \begin{align} \zeta(s) &= ...
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3answers
737 views

$1 + 1 + 1 +\cdots = -\frac{1}{2}$

The formal series $$ \sum_{n=1}^\infty 1 = 1+1+1+\dots=-\frac{1}{2} $$ comes from the analytical continuation of the Riemann zeta function $\zeta (s)$ at $s=0$ and it is used in String Theory. I am ...
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1answer
56 views

What is the number $A$? And the function $G$?

In one of my screenshots I found the following equality : ...
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1answer
63 views

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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3answers
179 views

Hard Definite integral involving the Zeta function

Prove that: $$\displaystyle \int_{0}^{1}\frac{1-x}{1-x^{6}}{\ln^4{x}} \ {dx} = \frac{16{{\pi}^{5}}}{243\sqrt[]{{3}}}+\frac{605\zeta(5)}{54} $$ I was able to simplify it a bit by substituting ${y = ...
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113 views

Challenging Infinite summation involving the zeta function [duplicate]

Evaluate: $$\large\sum_{k=1}^{\infty}\left(\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}\right)^2$$ MY ATTEMPT: Recognizing that $\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}$ can be written as $ ...
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3answers
723 views

How to prove that $\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$

Prove that $$\sum_{n=1}^{\infty}\frac{(H_n)^2}{n^3}=\frac{7}{2}\zeta(5)-\zeta(2)\zeta(3)$$ $H_n$ denotes the harmonic numbers.
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1answer
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Proof that $\zeta'(-2n)=(-1)^n\frac{(2n)!}{2(2\pi)^{2n}}\zeta(2n+1)$.

How would one prove the following statement which I found here, and/or does anyone know of a reference with a proof? $$\zeta'(-2n)=(-1)^n\frac{(2n)!}{2(2\pi)^{2n}}\zeta(2n+1).$$
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How to show that the Laurent series of the Riemann Zeta function has $\gamma$ as its constant term?

I mean the Laurent series at $s=1$. I want to do it by proving $\displaystyle \int_0^\infty \frac{2t}{(t^2+1)(e^{\pi t}+1)} dt = \ln 2 - \gamma$, based on the integral formula given in Wikipedia. ...
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2answers
105 views

Is it true that $\int_0^1 \lfloor x^{-1} \rfloor^{-1} x^n dx = \frac{1}{n+1}(\zeta(2)+\zeta(3) + \dots + \zeta(n+2) ) - 1$?

This question is inspired by the formula $$\displaystyle\int_0^1{\left\lfloor{1\over x}\right\rfloor}^{-1}\!\!dx={1\over2^2}+{1\over3^2}+{1\over4^2}+\cdots = \zeta(2)-1,$$ see for instance this ...
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Upper and lower bounds for the smallest zero of a function

The function $G_m(x)$ is what I encountered during my search for approximates of Riemann $\zeta$ function: $$f_n(x)=n^2 x\left(2\pi n^2 x-3 \right)\exp\left(-\pi n^2 x\right)\text{, ...
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0answers
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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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0answers
50 views

Intuitive explanation for $\zeta (2)=\frac{\pi^2}{6}$ [duplicate]

Using $f(x)=x^2$ Fourier' series, the proof for $\zeta (2)=\frac{\pi^2}{6}$ is pretty straight forward. I'm wondering if there is a more intuitive explanation for the equality, one that a layman could ...
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1answer
41 views

Convergence of $\sum_{n=1}^{\infty} n$ and integral test [duplicate]

I have read online, that one can show that $\sum_{n=1}^{\infty}n = -\frac{1}{12}$. But isn't this a Riemann Series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p=-1$. And if so, can't ...
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2answers
178 views

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} ...