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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16
votes
2answers
468 views

Simpler zeta zeros

Is it true that $$\lim_{y\rightarrow\infty}\dfrac{\sum_{n=1}^{y}n^{-1/2-iy}}{\zeta(1/2+iy)}=1$$ ? Below is a plot of $$\sum_{n=1}^{y}\dfrac{1}{n^{s}}\text{for }s=\dfrac{1}{2}+iy$$ set against its ...
5
votes
0answers
144 views

Log Log Integrals III

The integrals \begin{align} I_{7} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( \ln \left(\frac{1}{x}\right) \right) \ \frac{dx}{1-x} \end{align} and \begin{align} I_{8} = \int_{0}^{1} \ln(x) \ \ln^{2}\left( ...
1
vote
0answers
88 views

Linear combination of numbers: Express the $n^{th}$ Fibonacci number in terms of known constants.

Given the concept that any number can be expressed as a combination of other numbers can the $n^{th}$ Fibonacci number be expressed in terms of $\zeta(3)$ and $\ln(2/e)$ ? If possible, show all work ...
6
votes
1answer
97 views

Evaluation of $\int_{0}^{1} (1-x)^{-1/2} \ \ln^{2}(x) \ \ln^{2}(1-x) \ \mathrm{d}x$

Can the integral \begin{align} \int_{0}^{1} \frac {\ln^{2}(x)~\ln^{2}(1-x)}{\sqrt{1-x}}~\mathrm{d}x \end{align} be evaluated in terms of $\zeta(3)$ and $\ln(2/\mathrm{e})$ ?
1
vote
0answers
28 views

Periodicity in Riemann zeros.

Has someone studied if the non-trivial zeroes of the Riemann zeta function has some "periodicity" or "quasiperiodicity"? And what about generalized zeta functions and/or L-functions?
2
votes
1answer
100 views

GCD function and relation between Hurwitz and Riemann zeta function

Does anyone know how to show the following: $\sum_{k=1}^n\gcd(n,k)\zeta(s,\frac{k}{n})=\left(\sum_{k=1}^n\gcd(n,k)\right)\zeta(s)$
0
votes
0answers
17 views

Hurwitz Zeta Function and Dirichlet Characters

Is there a way to obtain a closed form solution to the sequence of summations: $ \sum_{m=1}^{k} \chi (m) \zeta(s,\frac{m}{k}) $ where $\chi (m)$ is a Dirichlet character modulo $k$ and ...
2
votes
3answers
104 views

$\zeta(2 + it) = \zeta(2-it)$

Let $\zeta(s)$ denote the Riemann zeta-function. Show that $\zeta(2 + it) = \zeta(2-it)$ for all real t. Give some hints how to do this one.Thanks in advance.
0
votes
0answers
18 views

Relation between zeta function and number of zeros of a homogeneous polynomial

For $f$ a homogeneous polynomial (i.e., every monomial has the same degree), $K$ a field of $q$ elements, one defines the zeta function $Z_f(u)$ as $$ Z_f(u)=\exp(\sum_{s=1}^\infty N_su^s/s) $$ ...
1
vote
1answer
43 views

Zeta Regularization and Products of Primes

How can one prove that: $2 * 3 * 5 * 7 \ldots = \prod_{n=1}^{\infty} p_i = 4\pi^2$ using zeta regularization? The sum diverges like the Ramanujan/Euler product but it can be associated to a value ...
4
votes
1answer
89 views

On the sum of prime powers

Has anybody investigated the asymptotic growth rate of functions in the form of $$f(z,n)=\sum\limits_{p\le n}p^z$$ For $Re(z)\ge -1$. Of course $f(0,n)=\pi (n)$ has an ocean of research surrounding ...
4
votes
2answers
259 views

Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
7
votes
1answer
140 views

Calculating $\pi$ via the $\zeta$ function?

I was fooling around, trying to come up with a rapid way to compute $\pi$. Then I remembered that we always have: \begin{equation} \zeta(2n)=c\pi^{2n}, \end{equation} where $n$ is a positive integer ...
5
votes
1answer
82 views

Question about $ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) \forall s\in \Bbb N$

what I found from messing around was $$ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) $$ $$ s\in \mathbb{N} $$ when the partial sum is changed to an equivalent polynomial using Faulhaber's formula. ...
0
votes
0answers
17 views

When is the fourier transform of a quasi-character $\hat c(\alpha)=|\alpha|c^{-1}(\alpha)$?

This is from lemma $2.4.2$ of Tate's thesis. Let $c$ be a quasi-character on $k^{*}$, the multiplicative group of a number field completed at a non-archimedian place. Lemma 2.4.2 For $c$ in the ...
2
votes
0answers
41 views

Relations betweens Multizeta Values

I am studying Multizeta values at the moment and I found that at weight 5, the basis is given by $\zeta(5)$ and $\zeta(3)\zeta(2)$ in the literature. Solving all shuffle and stuffle relations using ...
4
votes
2answers
182 views

Zeta function for negative integers

I already proved that $\zeta(z)=\frac{1}{\Gamma(z)}\int_0^\infty\frac{t^{z-1}}{e^t-1}dt=\frac{\Gamma(z-1)}{2\pi i}\int_{-\infty}^0\frac{t^{z-1}}{e^{-t}-1}dt$ Now the Benoulli numbers are defined by ...
0
votes
1answer
79 views

How to solve this summation (Lerch Transcendent)?

How is it possible to deduce the closed form of the following? $$\sum_{i = 0}^{n - 1} \frac{2^i}{n - i} = ?$$
2
votes
1answer
61 views

Evaluating Dirichlet series

It is well known that $$\eta(s)=\sum\limits_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^s} =(1-2^{1-s})\zeta(s)$$ But I have the wider problem of evaluating the following ...
1
vote
0answers
50 views
-3
votes
2answers
153 views

If the set of natural numbers is closed under addition, how can we have the result that the sum of all the natural numbers to infinity is -1/12 [duplicate]

As seen here and on this wikipedia page the sum of all the natural numbers to infinity is -1/12. $\sum_{n=1}^\infty n = \frac{-1}{12}$ but the set of natural numbers is closed under addition and ...
4
votes
1answer
79 views

Moebius / Zeta function connections

Following on from this question, I include a plot of the slightly less clear, but far simpler mathematically Mertens function against $x$ to the power of Zeta Zero 1, where the correlation between the ...
4
votes
0answers
54 views

Asymptotics for Mertens function

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: ...
4
votes
0answers
63 views

prime zeta function when $0<s<1$ [duplicate]

I would like to know if there is a good estimate for the sum which concerns all primes not exceeding $x$: $$\sum\limits_{p\leq x}\frac{1}{p^s}$$$$0<s<1$$. Only this. Thanks in advance!
2
votes
0answers
24 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
99 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
101 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
164 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
27 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
73 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
59 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
39 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
76 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
47 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 ...
32
votes
0answers
567 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
223 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
103 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
494 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
51 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
82 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. ...
12
votes
1answer
190 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
78 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
119 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
79 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
171 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
73 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
97 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
157 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 ...
27
votes
0answers
795 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
50 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 ...