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

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10
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2answers
266 views

Proving a formula related with zeta function

Could you show me how to prove the following formula?$$\sum_{n=1}^\infty\frac{\zeta (2n)}{2n(2n+1)2^{2n}}=\frac12\left(\log \pi-1\right).$$ In the 18th century, Leonhard Euler proved the following ...
1
vote
1answer
360 views

Riemann Zeta Function

Can somebody provide me with the formula for the sum of reciprocal of the roots of the Riemann zeta function? Also if $a+ib$ is a root, will $a-ib$ also be a root?
2
votes
2answers
288 views

New tools for complex analysis and application to the Riemann Zeta function?

I've worked as a graphic artist for the past fifteen years, thus I have no relationship with the academic mathematical community. It is therefore difficult for me to check some results. 1. Tools for ...
3
votes
1answer
981 views

Real and imaginary part of Gamma function

Is there a way to separate the real and imgainary part of the gamma function $$\Gamma (a+ib)$$ I thought of using the formula $$\zeta(z) \Gamma(z) = \int^{\infty}_0\frac{t^{z-1}}{e^t-1}\, dt$$ ...
2
votes
0answers
71 views

Validity of a functional formula of the Riemann Zeta function across the whole complex plane?

Could someone confirm me the validity of the following formula: $$\zeta\left(z\right)=2^{z}\pi^{z-1}\sin\left(\frac{\pi z}{2}\right)\left(\frac{e^{-\gamma ...
1
vote
1answer
86 views

Relation between $1-(n^{p-1}\mod p)$ and Riemann $\zeta$

Taking: $$\mathcal V_p=1-(n^{p-1}\mod p)$$ with $$\lim_{p\rightarrow \infty}\mathcal V_p = \operatorname{sinc}(2\pi \,n)$$ and Riemanns' well known functional equation, I get easily to this result: ...
3
votes
1answer
262 views

Proving that $\sum (-1)^{n+1} n^{-z}$ defines an analytic function in $Re z>0$

I want to show that the series $\sum_{n=1}^\infty (-1)^{n+1} n^{-z}$ converges to an analytic function for $\Re z>0$. For $\Re z>1$ the terms are dominated by $n^{-x}$ so that we have absolute ...
0
votes
2answers
64 views

What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$?

Let $s$ be complex and $\zeta(s)$ the Riemann Zeta function. What is a series expansion for $\zeta(s)$ valid iff $\operatorname{Re}(s)>\frac{1}{2}$ ? I want a series expansion such that ...
1
vote
3answers
116 views

This Riemann zeta function limit is always zero for any k, right?

I am quite certain that this limit: $$\lim_{s\to 1} \, \zeta (s) \prod _{n=1}^k \left(1-\frac{1}{n^{s-1}}\right)=0$$ is always zero for any integer $k \geq 2$. Can you prove it? I can't. I have to ...
16
votes
1answer
613 views

“Orientation” of $\zeta$ zeroes on the critical line.

I am pretty ignorant about complex analysis so please forgive my lack of terminology. I saw a pretty picture (posted below) of the behavior of the Riemann zeta function along the critical line. What ...
0
votes
1answer
77 views

To which extent distribution of Riemann non-trivial zeros follow a gauss process?

I am trying to clearer and preciser understand to which extent the distribution of the non-trivial zeros of the Riemann $\zeta$-function follow a Gauss process? Yet, what I figured out from ...
5
votes
2answers
179 views

Limit involving the Riemann zeta function, why is this identity trivial?

Mathematica knows that: $$n^k=\lim_{s\to 1} \, \frac{\zeta (s) \left(1-\frac{1}{\exp ^{s^{n^k}-1}(n)}\right)}{n}$$ Why is the above a trivial identity? What is it about the Zeta function that makes ...
8
votes
3answers
555 views

Riemann Zeta function - number of zeros

I want to write a program that calculates the number of zeros (It is not necessary to identify them, just the number of them) between 0 and x for the Riemann Zeta function, being x the imaginary part ...
9
votes
4answers
6k views

What is the analytic continuation of the Riemann Zeta Function

I am told that when computing the zeroes one does not use the normal definition of the rieman zeta function but an altogether different one that obeys the same functional relation. What is this other ...
8
votes
2answers
405 views

Tying some pieces regarding the Zeta Function and the Prime Number Theorem together

I came across two remarks that I would appreciate help in making the connections. I) In Riemann's Explicit Formula: for $x > 1$ $\Pi = Li(x) - \sum_{\rho:\zeta(\rho)=0}Li (x^{\rho})- \log(2) +$ ...
5
votes
1answer
189 views

Expanding Riemann Zeta

Consider the Riemann Zeta Function $\zeta(x) = 1 + 2^{-x} + 3^{-x} + 4^{-x}...$ Notice the following identity: $a^{-x} = (e^{ln(a)})^{-x} = e^{-xln(a)}$ Therefore: $\zeta(x) = 1 + 2^{-x} + 3^{-x} ...
1
vote
0answers
91 views

Would be this formula valid ? Zeta regularization and Euler product plus zeros.

$$ \frac{\zeta (0)}{\zeta (s)}= \prod _{p}\prod_{m= -\infty}^{\infty}\left(1-\frac{is\log p}{2\pi m}\right),$$ where $m$ does not run over $ m=0 $ and '$p$' means a product over all the primes :) ...
16
votes
1answer
399 views

What is a zeta function?

In my readings, I've come across a wide variety of objects called zeta functions. For example, the Ihara zeta function, Igusa local zeta function, Hasse-Weil zeta function, etc. My question is simple: ...
0
votes
1answer
119 views

Proof of a Dirichlet's theorem using the Riemann zeta function?

Someone could tell me if there is a proof of the Dirichlet's theorem on arithmetic progressions stated below using only the Riemann zeta function $\zeta(s)=\sum_{n=1}^\infty ...
3
votes
0answers
95 views

Integer values of the Riemann Zeta function

The when $s$ is real and greater than 1, the Riemann zeta function $\zeta(s)$ takes all finite positive value $> 1$. I am studying the values of $s$ for which $\zeta(s)$ is a positive integer. I ...
3
votes
0answers
144 views

question about riemann zeta function

How can one prove that $$\zeta (2n)=\frac{(-1)^{n-1}2^{2n-1}\pi ^{2n}B_{2n}}{(2n)!}$$ where $n\in N$ and how can one prove that $$\zeta (2n)=\frac{(-1)^{n}2^{2n-2}\pi ...
3
votes
1answer
71 views

How to show that this function (related to the zeta function) is even?

Consider the function $\phi(u) = 2 \sum_{n=1}^\infty (2n^4 \pi^2 e^{9u/2} - 3n^2 \pi e^{5u/2}) e^{-n^2\pi e^{2u}}$. This appears in Titchmarsh's "The Theory of The Riemann Zeta Function." On page ...
5
votes
0answers
122 views

Inverting the Riemann zeta function in $s>1$

Let $s>1$ be a positive real and the Riemann zeta fucntion be defined for $s>1$ as $$ \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$ I am looking for an inversion formula for the zeta ...
0
votes
1answer
64 views

How to introduce an integer function into $\zeta$ function instead of $n$

I have a problem that I am struggling with since long and probably it is simple but I can not get through. So your help would be very welcome. Known that Riemann $\zeta$ function is defined as sum ...
2
votes
1answer
131 views

The relation of $\zeta$-function and $p^k$ for $Re(s) \le 1$?

The von Mangoldt function: $$\Lambda(n) = \begin{cases} \log p &; \mbox{if }n=p^k \mbox{ for some prime } p \mbox{ and integer } k \ge 1, \\ 0 &; \mbox{otherwise.} \end{cases}$$ establishes a ...
3
votes
2answers
166 views

Analytic continuation of Riemann Zeta funtion

I am reading about zeta function from book by Ingham. In that book the following theorem is given. I am unable to understand what does he mean by finite part of plane in the statement.
7
votes
1answer
230 views

Can exceptionally large primes be used to get information on the roots of $\zeta$?

Take a list $L$ of roots of the $\zeta$ function, like the ones provided by Andrew Odlyzko and plug it into the Prime Counting Function $\pi(x)$ given by $$ \pi(x) \approx \operatorname{R}(x^1) - ...
8
votes
4answers
237 views

Showing that $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$

I need prove the following: ($\zeta(s)$ is the Riemann zeta function) $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$ I really don't know, i have tried, but nothing for now.
3
votes
2answers
321 views

Zeta function and probability

I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function) But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the ...
1
vote
1answer
64 views

Consider $\int \frac{(-x)^{s-1}}{e^x-1} dx $

I lost in this proof of Riemann's paper: On the Number of Prime Numbers less than a Given Quantity. If one now considers the integral $$ \int \frac{(-x)^{s-1}}{e^x-1} dx $$ from $\infty$ to $\infty$ ...
1
vote
0answers
122 views

Euler's Basel problem continued… $\zeta(2n)$ expressed in terms of $sinc$?

I have to make a brief intro before comming to my question. To approach the famous Basel problem Euler starts with the $sinc$ function \begin{align}\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + ...
3
votes
1answer
153 views

Riemann zeta function at zero

Can the value of Riemann zeta function at 0, $\zeta(0)=-1/2$, be deduced from the identity $E(z)=E(1-z)$, where $$E(z)=\pi^{-z/2}\Gamma(z/2)\zeta(z)?$$
7
votes
1answer
159 views

A question related to Riemann zeta function

Does anyone know why the following statement is correct? Let $f(x)$ be the function whose value on the interval $m\pi<x<(m+1)\pi, m=0,1,2,\cdots$, is $(-1)^m\frac{\pi}{4}$. Let $0<s<1$. ...
7
votes
2answers
431 views

Generating functions and the Riemann Zeta Function

The generating function for the terms of the harmonic series: $\frac{1}{n}$ is $-\ln(1 - x)$. Does an ordinary generating function exist for the terms of the zeta function $\zeta(s) = ...
13
votes
1answer
396 views

Proof of $\sum_{n=1}^\infty \frac{1}{n^4 \binom{2n}{n}}=\frac{17\pi^4}{3240}$

Recently, I was able to prove that $$\sum_{n=1}^\infty \frac{1}{n \binom{2n}{n}}= \frac{\pi}{3\sqrt{3}}$$ $$\sum_{n=1}^\infty \frac{1}{n^2 \binom{2n}{n}}= \frac{\pi^2}{18}$$ But does anybody know ...
2
votes
0answers
144 views

The partial sum and partial product of $\zeta$function

Taking the partial sum of the $\zeta $ function: $$\zeta^H(s,k)=\sum_{n=0}^k \frac{1}{n^s}$$ and the partial product f the $\zeta $ function: $$\zeta^P(s,j)=\prod_{i=0}^j \frac{1}{1-p_i^{-s}}$$ I ...
19
votes
3answers
451 views

Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$

I'm trying to figure out how to evaluate the following: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx $$ I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies ...
4
votes
1answer
213 views

question about Riemann zeta $\zeta (0)$ [duplicate]

i know that $$\zeta (m)=\sum_{n=1}^\infty n^{-m}$$ so $$\zeta (0)=\sum_{n=1}^\infty n^0=1+1+1+1+1+1+\cdots=\infty $$ but actually $$\zeta (0)=-0.5$$ where is the wrong please help thanks ...
8
votes
1answer
81 views

product of zeta(k)

The product $\prod_{k\geq 2}\zeta(k)$ converges. Is the limit a known constant ? (This infinite product is involved in the estimation of the covolume of $SL_n(\mathbf Z)\backslash SL_n(\mathbf ...
13
votes
4answers
4k views

Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$ I found this result very ...
0
votes
4answers
134 views

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
1
vote
1answer
71 views

Is there a pair correlation function for primes?

Montgomery's pair correlation function for the non-trivial zeros of the Riemann $\zeta(s)$ function is defined via the term $$1- \left( \frac {\sin(\pi u)}{\pi u} \right)^2$$ Does anybody know if ...
9
votes
1answer
710 views

Apéry's constant ($\zeta(3)$) value

I tried to find some proofs about the Apéry's constant, but I didn't find any intuitive proof. Is this constant given by the "brutal force" summing of $1 + \frac{1}{2^3} + \frac{1}{3^3} + ...
6
votes
1answer
144 views

Interesting phenomenon with the $\zeta(3)$ series

I noticed that if one takes certain partial sums of the series for $\zeta(3)$: $$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$ an interesting phenomenon occurs ...
13
votes
1answer
1k views

Zeta function zeros and analytic continuation

I'm learning about the zeta function and already discovered the intuitive proof of the Euler product and the Basel problem proof. I want to learn how to calculate the first zero of the Riemman Zeta ...
1
vote
2answers
173 views

Proof that the zeta function converges for Re(s)>1

It would be absolutely fantastic if anybody could give me some guidance on the question above. For me (please correct me if I'm wrong), this question boils down to proving that ...
7
votes
0answers
248 views

Identity involving $\zeta(3)$

This is related to this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that ...
2
votes
0answers
89 views

A function that generates 'alternating' non-trivial zeros of $\zeta(s)$

I am trying to find a function, that assuming RH, generates subsequent non-trivial zeros $\rho_n$ in an alternating way i.e.: $$\frac12+14.134...i,\frac12-21.022...i,\frac12+25.010...i, \dots$$ or ...
7
votes
3answers
300 views

What is the half-derivative of zeta at $s=0$ (and how to compute it)?

[Update 3:] I gave a new partial answer following the ansatz in question Q3. I leave the other parts of the question untouched, they are also partially answered in specialized other questions in MSE. ...
1
vote
1answer
116 views

Reformulation of riemann zeta

Does this extend to $\mathbb{C}$? $\displaystyle ζ(x) = \int_0^{\infty} \frac{ 1}{\lfloor t\rfloor ^x} dt$, where for $0 \leq t < 1$ we say that $\lfloor t \rfloor = 1$.