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

learn more… | top users | synonyms

0
votes
1answer
44 views

Does this limit involving the Dirichlet eta function and the Riemann zeta function make sense?

Let $p_n$ the sequence of prime numbers (and you will consider below, too, the sequence $\frac{1}{n}$ with $n>1$). And if it isn't wrong for $0<\Re s<1$ the known equation between Dirichlet ...
1
vote
1answer
45 views

What are the conditions on this Riemann-Zeta function functional equation?

I am a huge fan of the Riemann Zeta function's functional equation: $$\large{\color\green{\zeta(x)=2^x \Gamma(1-x)\zeta(1-x)\pi^{x-1}\sin\frac{\pi x}{2}}}$$ I am curious as to what conditions on $x$ ...
21
votes
2answers
412 views

Conjecture $\int_0^1\ln\ln\left(\frac{1+x}{1-x}\right)\frac{\ln x}{1-x^2}\,dx\stackrel?=\frac{\pi^2}{24}\,\ln\left(\frac{A^{36}}{16\,\pi^3}\right)$

I did some numeric experiments with integrals involving double logarithms (because they received much interest both on this site and in published papers, sometimes under names of ...
1
vote
0answers
36 views

How can I prove that $\zeta(2n) = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}$? [duplicate]

I tried to prove this with Valli's equation: $\frac{sinx}{x} = \prod(1-\frac{x^{2}}{(n\pi)^{2}})$ and use $\frac{d(ln(sin(x)))}{dx} = i + \frac{2i}{e^{2ix}-1}$. Maybe it's better to use Taylor's ...
0
votes
0answers
105 views

Fault in proof of $\zeta(2) = \frac{\pi^2}{6}$

Consider the proof of: $$\zeta(2) = \frac{\pi^{2}}{6}$$ So the proof assume that (because of Euler decomposition) $$\frac{\sin(x)}{x} = \prod_{n > 0}\left(1 - \frac{x^{2}}{(n\pi)^{2}}\right)$$ ...
6
votes
1answer
44 views

$L$-function absolutely convergent for $\text{Re}(s) > 1$, condition for $L(s, \chi)$ converging for $\text{Re}(s) > 0$?

I have two questions related to here. Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ...
6
votes
2answers
77 views

Product of two absolutely convergent Dirichlet series

We have$$(f * g)(n) = \sum_{d \mid n} f(d)g(n/d).$$How do I see that if the two Dirichlet series$$F(s) = \sum_{n =1}^\infty f(n)n^{-s},\text{ }G(s) = \sum_{n=1}^\infty g(n)n^{-s}$$converge absolutely ...
7
votes
1answer
135 views

Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $?

The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value. In general the logarithmic deriative of ...
1
vote
1answer
52 views

Can be justified this formula for $\zeta(2n+1)$

Can be justified for integers $n\geq 1$ that $$\zeta(2n+1)=\prod_{\text{p, prime}}\frac{1-\sigma(p^{2n})^{-1}}{1-p^{-2n}}?$$ Truly I don't know if I am wrong another time, when I use for an integer ...
0
votes
0answers
23 views

Residue of $\frac{\text{cot}(\pi z)}{z^6}$ at $0$

I am trying to compute $\zeta(6)$ = $\sum_1^{\infty} \frac{1}{n^6}$; I generally know how to do this using a residue-based proof, but I am stuck at the last bit, namely calculating the residue of ...
0
votes
1answer
40 views

Books on Zeta Regularization Product

Does anybody know some book on zeta regularization, and the zeta regularization product? I'm quite interested on the topic but I would need a book with some review...
0
votes
0answers
24 views

How to prove an equation involving $\sum_{p,n}\frac{\log (p)}{p^{n/2}}\delta_{\log (p^n)}(y)$

My context is that I could assume as true statements (I say this since claims about the type of convergence could be difficult to me) the first equation in [1], that could be written as the derivative ...
1
vote
0answers
52 views

Something fishy in the Zeta function

Recently I came across the Riemmann representation of the Zeta function as follows: $$\zeta (s) = (2^s)(\pi^s-1) \sin(\frac {\pi s} 2) \Gamma(1-s) \zeta(1-s) .$$ Now, I went ahead to calculate the ...
1
vote
0answers
58 views

Is it possible to compute $\Gamma(\rho_k)$ or $\zeta(3+2\omega_k)$, when $\rho_k=\frac{1}{2}+i\omega_k$ is a nontrivial zero of Riemann zeta function?

I believe that I can to use the equation (1) (from reference [1]), a well known equation involving the Gamma function, Riemann zeta function and the theta function $\psi(x)=\sum_{n=1}^\infty ...
2
votes
0answers
57 views

Why do you need to prove the error term goes to zero for the complete derivation of the Euler Product Formula?

I am doing a project on the Riemann-Zeta Function which begins by examining the Euler Product Formula. I understand the proof up until the point where it is made 'rigorous'. In other words, I ...
0
votes
0answers
27 views

What is the best (today) value of $c_k$ in $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \le c_k Y^k(t)$

From the Vinogradov-Korobov estimate, we have $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \ll Y^k(t)$ What is the best (today) value of $c_k$ in $\left|\frac{\zeta^{(k)}}{\zeta} (1+it)\right| \le ...
4
votes
3answers
225 views

How to solve this integral: $\int_{-\infty}^\infty\frac{x^2 e^x}{(e^x+1)^2}\:dx$

I am trying to solve an integral like this: $$ I=\int \frac{x^2 e^x}{(e^x+1)^2} dx $$ And I get this answer: $$ \int \frac{x^2 ...
1
vote
1answer
84 views

Some doubts about easy computations involving nontrivial zeros of Riemann's zeta function

On assumption of Riemann hypothesis when I write a complex zero (nontrivial zero) of zeta function as $\rho=\frac{1}{2}+it_\rho$, and I write $x^\rho$ as $\sqrt{x}e^{it_\rho \log x}$, then multiplying ...
1
vote
1answer
51 views

How to calculate non-trivial zeros of the Riemann zeta function

I wanted to know how riemann calculated some non-trivial zero of the zeta function. Would I like a manual calculation.
3
votes
2answers
110 views

Riemann Zeta Function integral

I was reading about the Riemann Zeta Function when they mentioned the contour integral $$\int_{+\infty}^{+\infty}\frac{(-x)^{s-1}}{e^x - 1} dx$$ where the path of integration "begins at $+\infty$, ...
2
votes
1answer
138 views

Proof of Riemann Hypothesis

This proof was released this year: http://arxiv.org/abs/1508.00533 Where is the mistake? I just found it and was wondering how obviously wrong it is.
0
votes
1answer
139 views

Average number of square free divisors for $n\leq x$

Let $d_{sf}(n)$ be the number of square-free divisors of $n$, and let $D_{sf}(k)=\sum_{n=1}^{k} d_{sf}(n)$ denote the corresponding summation function. Mertens showed that the asymptotic expansion of ...
5
votes
0answers
95 views

On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
2
votes
1answer
62 views

What is the effective lower bound on gaps between zeta zeros?

In this question here: Upper bound on differences of consecutive zeta zeros by Charles it is said that: "There are many papers giving lower bounds to: $$\limsup_n\ \delta_n\frac{\log\gamma_n}{2\pi}$$ ...
2
votes
0answers
48 views

Easy computations from a representation for Riemann zeta function, and Riemann zeta zeroes on critical line

The following series for Riemann Zeta function converges for $\Re s>0$ $$\zeta(s)=\frac{1}{s-1}\sum_{n=1}^\infty(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}).$$ See for example this site or Wikipedia [1]. ...
1
vote
1answer
30 views

Breaking up integral representations by convergence

A known integral takes the form of $$\zeta(3)=\frac{1}{2}\int_{0}^{\infty} \frac{t^2}{e^t-1}dt$$ Through Wolfram part of the integral converges to $$\int_{0}^{\infty} \frac{t}{e^t-1}dt = ...
1
vote
0answers
34 views

Can you rewrite this function to be independent of the variable $p$?

A while ago Roger L. Bagula came up with the idea to study this function: ...
2
votes
3answers
126 views

Evaluating an infinite sum involving possibly hypergeometric terms

I was considering the following infinite sum $$ A(n) = \sum_{k=2}^{\infty}\left[\frac{(-1)^{k+n-1}}{k^n}(0k -1)(k-1)(2k-1)...((n-1)k-1) \right] $$ Some cases: $$ A(1) = ...
8
votes
0answers
125 views

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
0
votes
1answer
62 views

Is this plot of the argument of the Riemann zeta function around ZetaZero(127) correct?

(*Mathematica 8 start*) Clear[n, k, t, z, FL, NZ] N[ZetaZero[127]] NZ[t_] = Arg[Zeta[1/2 + I*t]]/Pi; Plot[NZ[t], {t, 280, 284}] Plot[NZ[t], {t, 282.3, 282.6}] ...
2
votes
1answer
53 views

Generating function of Riemann zeta function

I want to know about the generating function of the Riemann zeta function which is related with the Laurent expansion at $z=0$. $f(z) := \dfrac{d}{dz} \log(\sin\pi z)$ $f(z) = \dfrac{1}{z} -2\sum ...
0
votes
0answers
47 views

$\sum \zeta (s)$ converging proof?

I was pondering about the fact that the Zeta function can be represented as an infinite sum, $\zeta (s) = 1/1^s + 1/2^s + 1/3^s +\ ...$, and I thought about the infinite sum of the zeta function ...
2
votes
1answer
56 views

An algebraic manipulation of the Zeta function

Consider the following form of the Riemann Zeta function: $s\in\mathbb{C}$, such that: $\left |s \right | > 1$ $\zeta \left ( s \right )= 1+2^{-s}+3^{-s}+4^{-s}+5^{-s}...$ Now, due to the ...
1
vote
1answer
58 views

Could someone explain how the Gram series relates to Riemann's function?

I was reading an article on the distribution primes which mentions the following equation for Riemann's function $R(x)$: $$R(x) = \sum_{n=1}^{\infty}\frac{\mu(n)}{n}\text{li}(x^{1/n}) = 1 + ...
1
vote
1answer
56 views

How to calculate $\zeta(s)$ using residues?

We have the Riemann zeta function $$ \zeta(s) = \frac{\pi}{2} \int_{\gamma} \frac{dz}{2\pi i} \, \frac{1}{z^s} \cot(\pi z) $$ and I want to use the Residue theorem in order to understand why it is ...
3
votes
1answer
38 views

asymptotic expansion in powers of $ 1/s $

How could I get the coefficients in the Dirichlet series expansion $$ \frac{\zeta ' (s)}{\zeta(s)}= \sum_{n=1}^{\infty} \frac{a(n)}{n^s} $$ for the logarithmic derivative of the Riemann zeta ...
0
votes
0answers
38 views

Problem inside the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$

I am having trouble with one step for the derivation of $\zeta(-k)= -\frac{B_{k+1}}{k+1}$ found here. In the below steps, how do we get from $$\frac{1}{\pi{i}}\Bigl(G(z)-2G(2z)\Bigr) = -F(z)+F(-z)$$ ...
10
votes
2answers
332 views

Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it. $\textbf{Background}$ I am an undergraduate student and I'm very interested in number theory. I've tried ...
4
votes
1answer
61 views

Riemann zeta function and the volume of the unit $n$-ball

The volume of a unit $n$-dimensional ball (in Euclidean space) is $$V_n = \frac{\pi^{n/2}}{\frac{n}{2}\Gamma(\frac{n}{2})}$$ The completed Riemann zeta function, or Riemann xi function, is $$\xi(s) ...
0
votes
0answers
18 views

Importance of the real-rooted asympototics of $f_n(z)$ that uniformly converges to Riemann $\Xi(z)$ function

We are learning Riemann $\Xi(z)$ and Riemann $\zeta(s)$ functions. This question is related to an earlier one. (1) Suppose that a family of functions, $f_n(z)$, uniformly converges to Riemann ...
12
votes
3answers
981 views

Are the nontrivial zeroes of the Riemann zeta function countable?

It is known that the set of non trivial zeros is an infinite set. But is it known if it is a countable, or uncountable infinite set?
1
vote
0answers
134 views

Assuming the Riemann hypothesis, does this integral give the Riemann zeta zeros when increasing Working Precision in Mathematica?

It is probably well known that the Riemann zeta zeros satisfy the following equation: $$\frac{\arg \left(\zeta \left(\rho _n+\frac{1}{1000000000000000}\right)\right)}{\pi }+\frac{\vartheta ...
2
votes
0answers
69 views

Is there a contiguous locus of the equality $a = b$ in $\zeta(\rho + \varepsilon)=a + î b$ in the near of a root?

This is just by an accidental couriosity: In the near of a root $\rho$ of the Riemann's zeta - can there be a continuous line starting from $\zeta(\rho)=0$ to $\zeta(\rho + \varepsilon_j)=a_j + î b_j ...
1
vote
1answer
196 views

Argument of the Riemann zeta function on Re(s)=1

I refer to the lovely answer to this question. Is there an exact formula for the argument of the Riemann zeta function? Specifically, I would like to know the arguments along the line Re$(s)=1$. Some ...
3
votes
2answers
61 views

Clausen zeta function

For $0 < \theta < 2\pi$, define $$\kappa(x,\theta) = \frac{1}{\zeta(x)}\sum_{n=1}^\infty \frac{e^{ in\theta}}{n^x}$$ for $\Re(x) > 1$. It is easy to see that $$\kappa(x,\theta) = ...
0
votes
0answers
27 views

Values of Riemann zeta function at small height along Re(s)=1

I am looking for sources which give numerical and theoretical computations of $\zeta(1+it)$ and the completed zeta function (without the $s(s-1)$) $$\xi(1+it)=\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)$$ ...
1
vote
0answers
94 views

When does linear combination of real-rooted entire functions of genus 0 or 1 remain real-rooted?

In our search of a family of entire functions to approximate Riemann $\Xi(z)$ function, we encounter the following family of functions: $$f_m(z,n,b)=\sum_{k=1}^m (-1)^k u_k(z,n,b)\tag{1}$$ where ...
4
votes
1answer
75 views

On evaluating the Riemann zeta function, including that $\zeta(2)\gt \varphi$ where $\varphi$ is the golden ratio

A week ago, I got the following : For a positive integer $k$, using Cauchy–Schwarz inequality, $$\left(\sum_{n=1}^{\infty}\frac{1}{n^k(n+1)^k}\right)^2\lt ...
1
vote
1answer
39 views

what is the value of Chebyshev function at non-integer value?

What is the value of Chebyshev $\psi(x)$ function at non-integer values ? For example, what is the value of $\psi(3.56)$? I have seen, in same place, it seems that $$\psi(3.56)=\psi(3)$$ And in ...
0
votes
0answers
45 views

Summation continuation, over no common divisors

Recently i've thought of a new step in my summations. I've been mentioning them in my previous questions. And i know there are certain point were it doesn't work. But most of the time it does. ...