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

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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
416 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
394 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
140 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
449 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
3k 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
131 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
691 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} + ...
5
votes
1answer
141 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 ...
6
votes
0answers
246 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
297 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
113 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 $\lfloor t \rfloor = 1$.
11
votes
2answers
404 views

Clausen and Riemann zeta function

This is an exercise from the American Monthly Problems from last year. I would like prove two formulas: (1) $\int_0^{2\pi}\int_0^{2\pi}\log(3+2\cos(x)+2\cos(y)+2\cos(x-y)) dxdy=8\pi ...
1
vote
1answer
70 views

What does this limit indicate?

$$\lim_{x\rightarrow\infty} \zeta(x)-\zeta(x)^{-1}-\zeta(x)^2 = -1$$ What does this limit indicate?
2
votes
1answer
117 views

What's the lowest real $x$ such that $\zeta(x)$ converges?

It's easy to prove that$$\zeta(1)=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...$$ diverges, and $$\zeta(2)=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...$$ converges to $\frac{\pi^2}{6}$. Intuiting the ...
12
votes
5answers
389 views

Alternating sum of multiple zetas equals always 1?

This is more in the category of "recreational math"... I was playing with multiple zetas, in the notation of $\zeta(k),\zeta(k,k),\zeta(k,k,k),\ldots$ as given in wikipedia. Looking at the alternating ...
-2
votes
3answers
222 views

Quantum uncertainty can explain the Riemann Hypothesis?

In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of ...
0
votes
1answer
203 views

Can anyone provide a proof for this conjecture?

Theorem?: Let $n$ be a positive interger, $n>1$, then Riemann zeta function can be expressed in terms of a multiple integral which exhibits the following form: $$ \displaystyle ...
15
votes
1answer
370 views

A Tough Series $\sum_{k=1}^\infty \frac{\zeta(2k+1)-1}{k+1}=-\gamma+\log(2)$

I have done series with $\zeta(2k)$ and $\zeta(k)$, but I have no idea with this one: $$\sum_{k=1}^\infty \frac{\zeta(2k+1)-1}{k+1}=-\gamma+\log(2)$$ $\gamma$ is the Euler–Mascheroni Constant. This ...
8
votes
4answers
432 views

Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$

Prove that : $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)^3}=\frac{\pi^3}{32}.$$ I think this is known (see here), I appreciate any hint or link for the solution (or the full solution).
1
vote
1answer
53 views

The minimum of a function

Could anyone possibly give me any help with finding the minimum of this function? I believe the result to be $2\pi |n|$ from page 619 of this paper by W. G. C. Boyd. \begin{equation} ...
1
vote
1answer
204 views

How does one calculate the amount of time required for computation?

For example, to compute the zeroes of the Riemann zeta function using the Euler-Maclaurin summation method one has to do O(T) work. The Euler-Maclaurin summation method for zeta is given by $ ...
4
votes
2answers
176 views

Riemann Zeta Function By Hand

This may be a stupid question but is there a way to calculate Riemann's Zeta Function by hand exactly or can you only estimate it?
5
votes
1answer
471 views

The meaning of the Euler Formula for Zeta

Does anybody know about a "meaning" behind the Euler Formula, what does it really say about the primes? I know that it is in equation to the zeta function and also how it is derived, but cannot find ...
2
votes
1answer
232 views

1+2+3+4+… = -1/12 [duplicate]

Consider the zeta function $\zeta(s)= \sum \limits_{n=1}^{\infty} \frac{1}{n^s}$. It is established that $ \zeta(-1) = -\frac{1}{12}$. Reference (Equation 90) Then we have $ \zeta(-1) = \sum ...
1
vote
0answers
41 views

The sum of the reciprocals of fourth powers [duplicate]

This problem is an extension of the well known basel problem and involves finding the sum of 1 + 1/16 + 1/81 ... = 1/1^4 + 1/2^4 + 1/3^4 ... 1/n^4 where n tends to infinity Euler managed to prove ...
3
votes
1answer
186 views

Discontinuities of $\sum \frac{x^{\rho}}{\rho}$

H. Edwards in his book on the zeta function says that $\sum\frac{x^{\rho}}{\rho}$ converges conditionally "even when $\rho ,1-\rho$ are paired." I tried calculating some terms (n = 500 or so) and ...
5
votes
3answers
129 views

Two trivial questions about zeta function

I have two questions concerning the Riemann zeta function which are rather trivial so if anyone can give me the answers that would be nice, here is what I`m interested in: 1) In the equality ...
18
votes
1answer
363 views

References to integrals of the form $\int_{0}^{1} \left( \frac{1}{\log x}+\frac{1}{1-x} \right)^{m} \, dx$

While extending my calculation techniques, with aid of Mathematica, I found that \begin{align*} \int_{0}^{1}\left( \frac{1}{\log x} + \frac{1}{1-x} \right)^{3} \, dx &= -6 \zeta '(-1) ...
2
votes
2answers
184 views

Series expansion of $\zeta(s)$ using the derivatives of $\zeta(0)$

When playing with series expansions, I stumbled upon the following relation: $$\zeta(s) := \sum \limits_{n=0}^{\infty} \left( \zeta^{(n)}(0) \frac{s^n}{\Gamma(n+1)} \right)$$ that seems to hold for ...
3
votes
1answer
145 views

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

I'm trying to find to value of: $$ J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(x)\,dx $$ Here's what I've done: $$ ...
8
votes
2answers
353 views

How to find integral of $\int_0^\infty \frac{\ln ^2z} {1+z^2}\mathrm{d}z$?

How do I find the value of $$\int_{0}^{\infty} \frac{(\ln z)^2}{1+z^2}\mathrm{d}z$$ without using contour integration, - using the usual special functions, e.g., zeta/gamma/beta/etc. Thank you,
14
votes
2answers
410 views

Can the Basel problem be solved by Leibniz today?

It is well known that Leibniz derived the series $$\begin{align} \frac{\pi}{4}&=\sum_{i=0}^\infty \frac{(-1)^i}{2i+1},\tag{1} \end{align}$$ but apparently he did not prove that $$\begin{align} ...
1
vote
0answers
110 views

Efficient way for computation of derivatives of $f(x) = \zeta(1-x) + 1/x $ at integer x?

For some exercises with (divergent) summation of the Stieltjes constants I'm trying a formula, which involves derivatives of the $\zeta()$ -function at negative integers; perhaps better formulated as ...
8
votes
1answer
728 views

how to understand $\log\zeta(s)$ (Riemann zeta function)?

I know that if a function $f$ is analytic and has no zeros in a simple connected region, then we can define $\log{f}$ making it analytic in that region. Let's assume $Re(s)>1$. Is $\zeta(s)$ ...
3
votes
1answer
113 views

Calculating an almost Gamma integral

How would you proof that $$I:=\int_{0}^{\infty}\frac{z^{x-1}}{e^{z}+1}dz=\left(1-2^{1-x}\right)\Gamma(x)\zeta(x)$$ I can rewrite the integral as ...
17
votes
3answers
606 views

A series involves harmonic number

How do we get a closed form for $$\sum_{n=1}^\infty\frac{H_n}{(2n+1)^2}$$
7
votes
1answer
199 views

Serre's proof that zeta function is meromorphic

I try to understand the proof of Chap. VI, n° 3.1, Prop. 10 in Serre's "A course in arithmetic" (page 70). The goal is to prove that zeta-function can be written as \begin{align*} ...
4
votes
1answer
208 views

About Euler's formula for Apery number

Euler's formula. $$\zeta(3)=\frac{\pi^2}{7}\left(1-4\sum_{m\ge 1}\frac{\zeta(2m)}{(2m+1)(2m+2)2^{2m}}\right)$$ I saw this formula in Wikipedia a few months ago. I have searched about Euler's ...
9
votes
2answers
380 views

A tough series: $\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$, need help

I was doing a integral which ends up with a tough series part: $$\sum_{k=1}^{\infty}\frac{\zeta(2k)}{(k+1)(2k+1)}$$ Mathematica says $$\frac12$$ Which agrees with the anwer...Anyone know how to ...
3
votes
1answer
98 views

How to evaluate $\xi(0)$?

How do I evaluate $\xi(0)$ for the Riemann xi function? I know $\xi(0) = \xi(1)$ and $\xi(0) = \tfrac{1}{2} \cdot 0 \cdot (-1) \cdot \Gamma(0) \cdot \zeta(0)$ $\xi(1) = \tfrac{1}{2} \cdot 1 \cdot 0 ...
3
votes
2answers
167 views

Another improper integral

Show that : $$\int_0^1\frac{(\sin ^{-1}x)^2}{x}\text{d}x=\frac{\pi ^2\ln 2}{4}-\frac78\zeta(3)$$ This integral is in "irresistible integrals" on page 122. I can't prove this one.
8
votes
2answers
296 views

Intuitive explanation with rigorous details why zeta has infinitely many zeros?

I have seen a proof outline that $\zeta$ has infinitely many zeros on the critical line here but what I really want is: Simplest possible (least "magic") argument that explains why zeta has ...
8
votes
3answers
981 views

Why does zeta have infinitely many zeros in the critical strip?

I want a simple proof that $\zeta$ has infinitely many zeros in the critical strip. The function $$\xi(s) = \frac{1}{2} s (s-1) \pi^{\tfrac{s}{2}} \Gamma(\tfrac{s}{2})\zeta(s)$$ has exactly the ...