Questions on the various generalizations of the zeta function of Riemann. Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.

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2
votes
0answers
23 views

Computing leading coefficient of $\zeta_K(s)$ at $s=0$

I am trying to prove the result stated in this Wikipedia page that $$\lim_{s\to 0} s^{-r} \zeta_K(s) = -\frac{h_k\cdot R}{w_K}$$ The formulae I have are: $$\begin{aligned}\xi_K(s) ...
7
votes
1answer
93 views

Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ ...
1
vote
1answer
86 views

The alternating zeta function and functional equation

The Dirichlet eta function (the alternating zeta function) is given by $$η(s)=∑_{n=1}^{∞}(-1)ⁿ⁻¹/n^{s}$$ The functional equation for $η(s)$ is given by $$η(s)=ϕ(s)η(1-s)$$ where ...
2
votes
1answer
159 views

Show that $f$ is harmonic

Let us consider the function: $$ f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right] $$ My ...
1
vote
1answer
26 views

zeta function and probability divisible by k and choosing squares

I have some difficulty starting with this question: For $s>1$, set $\zeta(s)=\sum_{n=1}^\infty n^{-s}$, and define $p: \mathbb{N} \rightarrow [0,1]$ by $p(n)=\frac{n^{-s}}{\zeta(s)}$. One can ...
1
vote
1answer
57 views

Difference between Meromorphic and Analytic Continuation

We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is ...
0
votes
0answers
58 views

Differential equation to model a pseudo-random behavior?

In the paper http://arxiv.org/abs/1401.3620, "The zeros of the Riemann zeta-function and the transition from pseudo-random to harmonic behavior", the author built a function based on a finite amount ...
1
vote
0answers
34 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
4
votes
1answer
64 views

Properties of Dedekind zeta function

Suppose $K$ is a quadratic field and $a_K(n)$ denotes the number of ideals in the ring of integers of $K$ whose norm is equal to $n$. Then I need to show that $$\sum_{n\leq x} a_K(n)=O(x).$$ Clearly ...
2
votes
0answers
46 views

$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$ [duplicate]

How to Find $$\zeta(4)=\sum_ {k=1}^{\infty}{\frac{1}{k^4}}$$ the most basic way possible? I know it's $\pi^4/90$ but to arrive at this figure? Curious, because I need it to solve the integral ...
27
votes
0answers
495 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
1
vote
1answer
200 views

what the RH equivalent for Riemann prime formula $\Pi(x)$?

Question follow the one answered already, zeros about Riemann Zeta function and some L-function Let's me try my best to make it clear on what I am asking. In his 1859 paper "On the Number of Primes ...
4
votes
1answer
97 views

Dirichlet series experiment - computing the rational coefficient

Let consider the sequence of numbers $a_n = 0,1,-1,0,1,-1,0,1,-1, ...$ extended periodically ( so it has period $9$, $a_{n+10}=a_n$. In fact, this is a Dirichlet character $a_n = \chi_9(n)$ modulo 9. ...
23
votes
3answers
430 views

$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2)$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
2
votes
1answer
44 views

Residue of the (partial) Dedekind zeta-function

If $$\rho_\nu := res_{s=1} \zeta_K,\nu(s)=lim_{s\to 1}(s-1)\zeta_{K,\nu}(s)=\frac{2^r(2\pi)^s}{\omega_K |disc(\mathfrak(O)_K)|^{\frac{1}{2}}} \textrm{ (*)}$$ how can I follow that $$\rho := ...
2
votes
1answer
78 views

The multiplication formula for the Hurwitz/generalized Riemann zeta function

I'm having a difficult time showing that $$ \displaystyle \zeta(s,mz) = \frac{1}{m^{s}} \sum_{k=0}^{m-1} \zeta \left(s,z+\frac{k}{m} \right) $$ A couple of authors referred to it as an obvious fact. ...
2
votes
0answers
55 views

Divergent product, regularization [closed]

It is no secret that the Zeta function can do magic things. What we can do with this expression $$\prod_{k,m\in \mathbb Z}(k^2+m^2)$$ that it will be not infinity?
12
votes
1answer
184 views

Analytic continuation of Dirichlet function

Suppose $\{a_n\}$ is a sequence of complex numbers such that the sums $A_n=a_1+\cdots+a_n$ satisfy $$|A_n-nb|\leq Cn^{\sigma}$$ for all $n$, where $b\in\mathbb{C},C>0,0\leq\sigma<1$. Prove that ...
3
votes
0answers
72 views

How to get from Chebyshev to Ihara?

The number of returning paths on cubic graphs of length $r$ without backtracking may be written as $2^{-r/2}p_r(x/\sqrt{2})$ which is a Chebyshev Polynomial of the Second Kind $U_r(x)$. The linked ...
1
vote
1answer
116 views

proof regarding zeta function of a curve from Ireland and Rosen's “A Classical Introduction to Modern Number Theory”

In chapter $11$ section $5$ titled "the last entry" the authors state that the number of solutions to the congruence $x^2+y^2+x^2y^2 \equiv 1 \mod p$ is $p+1-2a$, where $p=a^2+b^2$ and $a+bi \equiv ...
2
votes
0answers
77 views

Zeta Zeros and primes / prime powers

The plot $\Re\ x^{Zeta\ Zero} + \Im\ x^{Zeta\ Zero}$ for the first $1000$ Zeta Zeros up to $x = 30$ using the following Mathematica code: ...
4
votes
1answer
167 views

Roots of some modified Bernoulli polynomials

Update The polynomials are generated as follows: Where $B_n(x) = \sum_{k=0}^n {n \choose k} b_{n-k} x^k$ is used to generate standard Bernoulli polynomials, top plot is generated as follows: ...
6
votes
1answer
71 views

Evaluating limit $\lim_{m\to{\infty}}\frac{\sum_{k=1}^m\cot^{2n+1}(\frac{k\pi}{2m+1})}{m^{2n+1}}$

How can I prove the following equality? $$\lim_{m\to{\infty}}\frac{\displaystyle\sum_{k=1}^m\cot^{2n+1}\left(\frac{k\pi}{2m+1}\right)}{m^{2n+1}}=\frac{2^{2n+1}\zeta(2n+1)}{\pi^{2n+1}}$$
7
votes
1answer
94 views

closed form of $\frac{(-\pi)^m\zeta(m,\frac{x}{\pi})+\pi^m\zeta(m,1-\frac{x}{\pi})}{(-1)^{m}\pi^{2m}}$

I know that following equality is true. $$\sum_{n=-\infty}^{\infty}\frac{1}{(x+n\pi)^m}=\frac{(-\pi)^m\zeta(m,\frac{x}{\pi})+\pi^m\zeta(m,1-\frac{x}{\pi})}{(-1)^{m}\pi^{2m}}$$ But can we find the ...
5
votes
1answer
152 views

Is $\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$ a rational number for $m,n\ge 2\in\mathbb N$?

Question : Is $$\frac{\zeta (m+n)}{\zeta (m)\zeta (n)}$$ a rational number for $m,n\ge 2\in\mathbb N$ where $\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$? Motivation : We know that $$\zeta ...
17
votes
0answers
568 views

The log gamma integral $\int_{0}^{z} \log \Gamma (x) \ \mathrm dx$

One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \ \mathrm dx $ is in terms of the Barnes G-function. $$ \int_{0}^{z} \log \Gamma(x) \ \mathrm dx = \frac{z}{2} \log (2 \pi) + ...
1
vote
1answer
47 views

A limit involving the Hurwitz/generalized Riemann zeta function

I want to show that $$ \lim_{s \to 1} \Big( \zeta(s,a) - \frac{1}{s-1} \Big) = - \psi(a)$$ where $\zeta(s,a)$ is the Hurwitz zeta function and $\psi(a)$ is the digamma function. The only thing I ...
5
votes
2answers
127 views

Looking for explanation of bound on Dirichlet's L-Function

I am reading Stein and Shakarchi's Fourier Analysis text and the proof Dirichlet's theorem and I am looking for clarification on how he derives the following for large $s$, $\lim_{s\to\infty}$ and ...
1
vote
0answers
22 views

What are Selberg class functions of degree two?

My question is: What are Selberg class functions of degree two. I know about the Selberg class. But I am not able to understand what means by degree two.
1
vote
0answers
29 views

How one can determine the rapidly convergente series?

The motivation to this question can be found in http://wstein.org/books/bsd/bsd.pdf In page 9 the author claimed that: 1.4.1. Approximating the Rank. Fix an elliptic curve $E$ over $Q$. The usual ...
1
vote
0answers
59 views

Prove that the only negative real zeroes are at the integers

Let $$L(C,s)=\prod_{p\mid\Delta}(1-a_{p}p^{-s})^{-1}\cdot\prod_{p\nmid\Delta}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse-Weil $L$-function of ...
1
vote
0answers
76 views

Conjectures about zeta functions and poles

Let $p^*_n$ be the $n$ th element of a subset of primes such that $p^*_{n+1}>p^*_n$ and $p^*_n < O((n+2) ln((n+2))^3)$. Define $f(z)$ as the analytic continuation of $\prod_{n>0} ...
4
votes
1answer
155 views

Mean density of the nontrivial zeros of the Riemann zeta function

As part of my MSc I am reviewing a paper. The paper is a review on the statistical distribution of the unfolded zeros (see below) of the Reimann functional equation. In the paper there is a sentence: ...
8
votes
0answers
249 views

Why are minima of $(k \bmod 4)$-Prime $\zeta$ functions $|P_x(r,t)|$ more frequent for $\frac\pi2\leq t \leq \pi$?

I got these plots when I evaluate the sum of truncated $(k \bmod 4)$-Prime $\zeta$ function, i.e. $$ P_x(r,t)=P_{x;4,1}(-ir\cos t)+P_{x;4,3}(-ir\sin t)=\sum_{x\geq p\;\bmod\;4=3} p^{-ir\cos ...
1
vote
0answers
124 views

The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
5
votes
1answer
89 views

Counting numbers of the form $ai + bj + cij$ and finding related L-series?

Let $a,b,c$ be given nonnegative integers with $gcd(a,b,c)=1$. Consider a given positive integer $n$ and positive integers $i,j$. Let $f_n(a,b,c)$ be the number of distinct solutions to $1<ai + bj ...
1
vote
1answer
34 views

how to prove that $\zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt$

how to prove that $$\zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt:R(s)>1 , R(q)>0$$ where $\zeta(s,q)$ is Hurwitz zeta function
10
votes
2answers
229 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 ...
0
votes
1answer
147 views

Infinite sum and equality between coefficients of the same index

I have two infinite sums that forms an equality: $$\sum_{n=1}^{\infty} \left(\zeta(2n)\frac{x^{2n}}{\pi^{2n}}\right) = \sum_{n=1}^\infty ...
4
votes
1answer
63 views

Pointwise convergence domain of function series

I'm stuck at finding pointwise convergence domain of the following function series $$\sum_{n=1}^\infty \frac{\sqrt[3]{(n+1)}-\sqrt[3]{n}}{n^x+1}$$ I tried to use d'Alembert and Weierstrass tests, but ...
5
votes
1answer
76 views

Where are the zeros of $\prod\limits_p (1-(p-1)^z)$?

Define $f(z)$ as the analytic continuation of $\prod\limits_p (1-(p-1)^z)$ where $z$ is complex and the product is over the odd primes $p$. Where are the zeros ($f(z)=0$) of this function ?
9
votes
2answers
190 views

Why does $\gamma=\lim_{s\to1^+}\sum_{n=1}^{\infty}\left(\frac{1}{n^s}-\frac{1}{s^n}\right)=\lim_{s\to0}\frac{\zeta(1+s)+\zeta(1-s)}{2}$?

To be clear, I'm having trouble with proving both equalities, and would appreciate a hint. I'm also not sure why $1^+$ must be used as opposed to $1^-$. I'm not sure about the definition of $\zeta(x), ...
7
votes
3answers
390 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 ...
3
votes
1answer
64 views

$L$-function of character in terms of other character

Let $F/K$ be a finite Galois field extension and $\varphi : \mathfrak{I}_K \rightarrow \mathbb{C}^{\times}$ a Hecke character of $K$.Define $\psi = \varphi \circ N_{F/K}$ as a Hecke character of $F$. ...
9
votes
1answer
198 views

L-function for Dirichlet characters vs Hecke character

For a Dirichlet character $\chi: \left(\mathbb{Z}/p\mathbb{Z}\right)^{\times} \to \mathbb{C}$, the Dirichlet L function is $$\prod_{q \neq p} (1 - \chi(q)q^{-s})^{-1}$$ If we lift this character to ...
2
votes
1answer
147 views

Recurrence relation in complex domain

I am bumping in the following problem : the expansion of $$x^{i/k} \operatorname{LerchPhi}[x,1,i/k]$$ leads, for each value of i, to a linear combination of k terms, each of them writing $$a(j) ...
0
votes
0answers
46 views

Gelfand Shilov vol 1. question: finite part of an integral

how do Gelfand and Shilov prove that the finite part $$ F.P \int_{0}^{\infty}x^{m}dx=0 $$ however i get for non zero lower limit the recurrence of integrals in terms of the Riemann zeta function $$ ...
14
votes
1answer
284 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: ...
53
votes
3answers
4k views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( ...
4
votes
1answer
258 views

Sum of Infinite Series with the Gamma Function

I am computing the volume of an infinite family of polytopes and have run into the following sum, which I am not sure how to evaluate, as it looks similar to the Riemann zeta function, except with the ...