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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5
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96 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 ...
5
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0answers
136 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( ...
5
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1answer
153 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 ...
5
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0answers
142 views

Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?

Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
5
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0answers
134 views

Shintani cone zeta function

Is there a procedure/algorithm for calculating sums of the form $$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$ where $$ L_i(n_1,\ldots, n_r) ...
4
votes
2answers
174 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 ...
4
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1answer
297 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 ...
4
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2answers
125 views

Challenging Infinite summation involving the zeta function [duplicate]

Evaluate: $$\large\sum_{k=1}^{\infty}\left(\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}\right)^2$$ MY ATTEMPT: Recognizing that $\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}$ can be written as $ ...
4
votes
2answers
242 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 ...
4
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1answer
65 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 ...
4
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1answer
348 views

Why do authors claim that Euler gave no proof to his “$\sin(\pi x)= \pi x\prod\limits_{k=1}^{\infty}\left(1-\frac{x^2}{k^2} \right )$” when…

When he proved the relation between $\pi \cot(\pi x)$ and the harmonic series in "Introductio in analysin infinitorum" which states that $$\pi \cot(\pi x)=\sum_{k \to \infty}^{\infty} ...
4
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1answer
101 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. ...
4
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1answer
176 views

Practical uses of the class number formula

For what number fields $K$ can we actually compute the residue of $\zeta_K(s)$ at the pole $s=1$ directly? Since the class number formula tells us that ...
4
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1answer
97 views

How to evaluate the following infinite sum involving the Riemann zeta function?

I want to find a closed form for the following infinite sum: $$\sum_{k=2}^{\infty} \frac{(-1)^k\cdot(k-1)}{k\cdot(k+1)}\cdot \zeta(k)$$ Is it possible? My approach was to transform it into a double ...
4
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1answer
108 views

Analytic Continuation of Zeta Function using Bernoulli Numbers

In my complex analysis textbook by Stein and Shakarchi, as an exercise, I am supposed to extend $\zeta(s)$ to the entire complex plane using Bernoulli numbers, but I am stuck. I can prove that $$ ...
4
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1answer
88 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
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1answer
169 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: ...
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1answer
26 views

coefficients of the zeta function of curve over a finite field $\mathbb{F}_q$

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
4
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1answer
78 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 ...
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51 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: ...
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62 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!
4
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1answer
71 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 ...
4
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1answer
164 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: ...
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145 views

When functions, analytically continued, carry over certain properties

Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
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3answers
337 views

Rankin-Selberg zeta function

I was reading this paper by de Weger and in conjecture 7 he mentions "the Riemann hypothesis for the Rankin-Selberg zeta function associated to the weight 3/2 modular form associated to E (an elliptic ...
3
votes
2answers
148 views

On the Hurwitz Zeta Function

In my mathematics course in Uni. (I'm a physics student) my prof. gave us the following exercise: to express the Hurwitz Zeta function $\zeta(2k+1,\frac{1}{4})$ with $k=1,2,3,\dots$ in terms of the ...
3
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1answer
78 views

Extending the zeta function to semiprimes, etc.

The Riemann Zeta function is defined for $s > 1$ as \begin{align} &\prod _{n=1}^{\infty}\dfrac{1}{1 -\ p_{n}^{\ \ -s}}\\ \end{align} It is possible to extend the zeta function to semiprimes ...
3
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1answer
47 views

Calculating zeta functions over a field

I am learning about zeta functions and have been trying the following example: Calculate the zata function of $x_0x_1-x_2x_3$ over $\mathbb{F}_p$. Does there exist an easy formula for calculating ...
3
votes
1answer
66 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 $F$. ...
3
votes
1answer
299 views

An example of divergent series with the Lerch function

I am often working with divergent series all around being this the bread and butter for a theoretical physicist. Thanks to the excellent work of Hardy these have lost their mystical Aurea and so, they ...
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0answers
24 views

Can derivative of Hurwitz Zeta be expressed in Hurwitz Zeta?

Can the derivative of Hurwitz Zeta function by the first argument be expressed in terms of Hurwitz Zeta and elementary fuctions? There is a formula which expresses Hurwitz Zeta through its ...
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0answers
76 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 ...
3
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0answers
96 views

Curious $\sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$ identity [duplicate]

Let $$F(x) = \sum _{n=1}^{\infty} \frac{1}{n^2 - x^2}$$ It seems that for odd integer $k$ $$F\left(\frac{k}{2}\right) = \frac{2}{k^2}$$ My evidence is strictly computational, and I have no idea how ...
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76 views

Can that two double series representations of the $\eta$/$\zeta$ function be converted into each other?

By an analysis of the matrix of Eulerian numbers(see pg 8) I came across the representation for the alternating Dirichlet series $\eta$: $$ \eta(s) = 2^{s-1} \sum_{c=0}^\infty \left( ...
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370 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_\color{red}x(\color{blue}s)=\sum_{p<\color{red}x} \frac{1}{p^{\color{blue}s}} =\sum_{\color{green}n=1}^{\infty}\frac{ \mu (\color{green}n)}{\color{green}n} ...
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0answers
150 views

How to get the floor function as a Mellin inverse of the Hadamard product of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely : $$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...
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0answers
105 views

Gram's series for integral equation

The prime counting function $ \pi(x) $ satisfies the integral equation $$ \log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$ and it has the solution in terms of Gram's ...
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0answers
178 views

Is this formula for $\zeta(15)$ true?

Apery gave, $\begin{aligned} \zeta(3) &= \frac{5}{2}\,\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k^3\,\binom {2k}k}\end{aligned}$ J. Borwein and D. Bradley found this can be generalized to ...
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0answers
255 views

Connection between Bernoulli polynomials and polygamma function

There is an intricate connection between Hurwitz Zeta and the (traditional) polygamma function: $$\psi_n(z)=(-1)^{n+1}n!\zeta(n+1,z)$$ If to use a generalization for Bernoulli numbers, this can be ...
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votes
3answers
254 views

Riemann Zeta formula

can anyone check if this formula is plausible ?? $$ \frac{1}{\zeta (s)} = \sum_{n=0}^{\infty}\frac{ (-\pi)^{n}(s-1)s}{2n!(s+2n)(s+2n+1)} $$ according to the authors this formula would be valid only ...
2
votes
3answers
98 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.
2
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1answer
254 views

Closed-form expressions for $\sum^N_{n}\frac{1}{n^2}$ ($n$, even or odd)

I am trying to find a closed-form expression for the finite sum $$\sum^N_{n=1}\frac{1}{n^2}$$ when $n$ is even and when $n$ is odd. I know that for $N\to\infty$ these series converge towards ...
2
votes
4answers
389 views

Generalization $\zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $

This is more a reference-request for some fiddling/exploration with the $\zeta$-function. In expressing the $\zeta$ and the alternating $\zeta$ (="$\eta$") in terms of matrixoperations I asked myself, ...
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3answers
106 views

Hint on a limit that involves the Hurwitz Zeta function

I will be honest. Some play with a weird integral has gotten me to this formulation: $$\lim\limits_{n\mathop\to\infty}\frac{\zeta(2,n)}{\frac 1n+\frac 1{2n^2}}=1$$ It seems true because of the ...
2
votes
1answer
148 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) ...
2
votes
1answer
207 views

What is this series called and when does it diverge?

What is this series called (if it has a name)? When does it diverge without analytic continuation and when does it diverge with analytic continuation? $\sum_{k_1,\dots,k_m=1}^{\infty} ...
2
votes
1answer
98 views

GCD function and relation between Hurwitz and Riemann zeta function

Does anyone know how to show the following: $\sum_{k=1}^n\gcd(n,k)\zeta(s,\frac{k}{n})=\left(\sum_{k=1}^n\gcd(n,k)\right)\zeta(s)$
2
votes
1answer
101 views

Elementary method to compute $\zeta (3)$

What is an elementary method to compute $\zeta (3)$ which can be understood by a high school student? I know how to compute $\zeta (2)$ but not $\zeta (3)$. Any help is very much appreciated.
2
votes
2answers
167 views

Estimating the integrated Tchebychev function and calculating its error

I would like to understand how to derive (2) from (1) below. Problem: If $\psi_1$ is the integrated Tchebychev function below $$\psi_1(x)=\frac{1}{2\pi i} ...
2
votes
1answer
261 views

Periodic Zeta Function Functional Equation

Recall that the periodic zeta function has the Dirichlet series $$F(\lambda,s)= \sum_{n=1}^\infty \frac{e^{2\pi i n\lambda}}{n^s}.$$ This defines an analytic function for $\Re s>0$ and has a ...