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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7
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1answer
113 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 \\ \...
7
votes
1answer
167 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 derivative:...
7
votes
0answers
223 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 ...
6
votes
3answers
117 views

Is $\sum_{n=1}^{\infty }\frac{8n\cdot\zeta (2n)}{3\cdot 2^{2n}}=\zeta (2)$?

$$\sum_{n=1}^{\infty }\frac{8n\cdot\zeta (2n)}{3\cdot 2^{2n}}=\zeta(2)$$ By using numerical calculation, I found this relationship between the values of zeta function at even integers and $\zeta(2)$,...
6
votes
1answer
408 views

Getting my number theoretic series straight

There are Artin $L$-series and Dirichlet $L$-series, and zeta functions for varieties and for number fields; there are a slew of objects named after Hecke... There are also various kinds of characters ...
6
votes
1answer
102 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 $\ln(2/\mathrm{e})$ ?
6
votes
1answer
104 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. ...
6
votes
1answer
383 views

Square-free zeta function zeros

It is a well known fact that the geometric series $$1+x+x^2+x^3+\ldots$$ has the following form $$\frac{1}{1-x}$$ Another possible representation is $$\prod_{k=0}^{\infty}\left(1+x^{2^{k}}\right)$$ ...
6
votes
1answer
299 views

On the Dirichlet beta function sum $\sum_{k=2}^\infty\Big[1-\beta(k) \Big]$

Given the Dirichlet beta function, $$\beta(k) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^k}$$ (The cases k = 2 is Catalan's constant.) It seems, $$\sum_{k=2}^\infty\Big[1-\beta(k) \Big] = \frac{1}{4}\...
6
votes
1answer
234 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 $$ \...
6
votes
0answers
56 views

Why was the zeta function introduced?

I know the 'Zeta Function' is very useful in Mathematics, and that it has relations with many other functions (such as the 'Gamma Function'). I also know the 'Zeta Function' $\zeta(s)$ is defined as: ...
6
votes
0answers
81 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
6
votes
1answer
99 views

Summation of a function

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}...p_k^{c_k}$ where $c_k = ...
6
votes
0answers
87 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ $$\...
6
votes
1answer
214 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 (2k)=(-1)^...
5
votes
1answer
131 views

Proving that $\sum\limits_{n=2}^{\infty }\frac{\zeta (n)}{2^{n-1}}=\log(4)$

$$\frac{\zeta (2)}{2}+\frac{\zeta (3)}{2^2}+\frac{\zeta (4)}{2^3}+\frac{\zeta (5)}{2^4}+...=\log(4)$$ I tried to prove it, but the problem with the odd zeta terms so that I don't have a function ...
5
votes
1answer
567 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 ...
5
votes
1answer
97 views

Cauchy-Ramanujan Formula $ \displaystyle \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} $

Cauchy and Ramanujan both gave the formula: $$ \sum_{\stackrel{m \in \mathbb{Z}}{m \neq 0}} \frac{\coth m \pi}{m^{4p+3}} = (2\pi)^{4p+3}\sum_{k=0}^{2p+2} (-1)^{k+1} \frac{B_{2k}}{(2k)!}\frac{B_{4(...
5
votes
3answers
887 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 ...
5
votes
1answer
193 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 $$\textrm{Res}_{s=1}\zeta_K(s)=\frac{2^r(2\pi)^...
5
votes
2answers
110 views

Generalizing $f(n)=\int_0^\infty \frac{1}{e^{x^n}+1}=\left(1-2^{(n-1)/n}\right )\zeta(n^{-1})\Gamma(1+n^{-1})$

I have come up with the following solution to this integral, but is just incomplete to my standards $$f(n)=\int_0^\infty \frac{1}{e^{x^n}+1}=\left(1-2^{(n-1)/n}\right )\zeta(n^{-1})\Gamma(1+n^{-1})$$ ...
5
votes
3answers
110 views

Sum involving zeta functions

Find closed form of the following - $$ \displaystyle \sum_{n=2}^{\infty}{\left(\frac{(n-1)\zeta(n)}{4n-1}\right)} $$ I don't know how to approach to it - Using the integral definition? I cannot use ...
5
votes
1answer
78 views

Is the relation $P(n) \sim \frac{1}{2^n}$ already known?

Apologies in advance if there is a violation of rules/laws here, as I am not a mathematician. $$ \begin{align} \lim_{n\to\infty} \left( \frac{\pi^{n}}{\zeta(n)}P(n) \right)^{\frac{1}{n}} &= \...
5
votes
2answers
150 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 $\...
5
votes
1answer
82 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 ?
5
votes
1answer
199 views

Zeta-like function with offset

Is there a known function of the form: $$\zeta(s,a) = \displaystyle\sum_{n=1}^\infty\frac{1}{n^s+a},$$ and if so what is its name?
5
votes
2answers
196 views

Lerch-$\small \zeta(\varphi,0,-n)$ of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $ is real) for $\small n \ge 2$?

Are the Lerch-$\zeta(\varphi,0,-n) $ of integer n (for shortness I use the notation of my earlier question $\small \zeta_\varphi(-n)$) periodically purely real and imaginary: $\zeta_\varphi (-n)^2 $ ...
5
votes
1answer
101 views

Mysterious Inverse Mellin transform using residue theorem

The origin of this problem lies in the explanation of the evaluation of the series $\sum_{n\geq1}\frac{\cos(nx)}{n^2}=\frac{x^2}{4}-\frac{2\pi}{4}+\frac{\pi^2}{6}$ see this link ( Series $\sum_{n=1}^...
5
votes
1answer
111 views

Log Log Integrals II

The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - \frac{\...
5
votes
1answer
223 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: $...
5
votes
1answer
108 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
votes
0answers
158 views

Can these integrals be represented in closed form?

This paper in the formula F.3.6 (page 271) gives the following formula for the derivative of Hurwitz Zeta function: $$\frac d{dz}\zeta(z,a)=-\frac12\ln(a)a^{-z}-\frac{\ln(a)a^{1-z}}{(z-1)^2}-\int_0^\...
5
votes
0answers
180 views

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
5
votes
0answers
167 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
votes
0answers
160 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
votes
0answers
148 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
4answers
167 views

How to show $\zeta (1+\frac{1}{n})\sim n$

How to show $\zeta (1+\frac{1}{n})\sim n$ as $n\rightarrow \infty$ where $\zeta$ is the Riemann zeta function. And can we say $\lceil \zeta (1+\frac{1}{n}) \rceil=n$ for any positive integer $n\geq 1$...
4
votes
2answers
112 views

Find: $\zeta \left( 3,1,1,1 \right)$

While solving a summation, I came across this: $$\zeta \left( 3,1,1,1 \right)=? $$ I'm new to multiple zeta values. That's why I couldn't find this. So my question is does a closed form exist ...
4
votes
3answers
89 views

Asymptotic formula for $\prod_{k=1}^{\infty}\zeta (2kn)$

Suppose $n\geq 1$ is a positive integer. Can we find an asymptotic formula for this product below. $$\prod_{k=1}^{\infty}\zeta (2kn)=\zeta (2n)\zeta (4n)\zeta (6n) \cdots$$ I tried to use $\zeta (2n)...
4
votes
3answers
231 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
votes
2answers
119 views

Proof of Functional Equation Zeta

$$ \pi^{-s/2}\zeta(s)\Gamma(s/2)=\pi^{-(1-s)/2}\zeta(1-s)\Gamma((1-s)/2) $$ (That's the equation that want to prove) Hello guys, so I'm trying to prove the functional equation of Riemann Zeta, ...
4
votes
2answers
437 views

What are all functions of the form $\frac{\cosh(\alpha x)}{\cosh x+c}$ self-reciprocal under Fourier transform?

There are some functions that are self reciprocal under cosine Fourier transform: \begin{equation} \frac{1}{\cosh x}, \frac{\cosh x}{\cosh 2x},\frac{1}{1+2\cosh x} \end{equation} It seems that they ...
4
votes
1answer
132 views

Any possible suspects for $\zeta(3)$?

I'm young, and have been studying this number for quite some time. Possible suspects for a closed form i have personally encountered through ghetto makeshift studyies are: Euler-Mascheroni Constant ...
4
votes
1answer
80 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
votes
1answer
415 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} \frac{1}{x+k}=\...
4
votes
1answer
46 views

Prove that this summation evaluates out to $\zeta(2)-1$

I am aware of the following identity: $$\sum_{m=1}^\infty \left(\frac{1}{m}-\left(\zeta(2)-\sum_{n=1}^m \frac{1}{n^2}\right)\right)=\zeta(2)-1$$ I can't quite figure out how to prove this result. ...
4
votes
1answer
130 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
votes
1answer
164 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
votes
1answer
194 views

Infinite Series $\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$ [duplicate]

Evaluate: $$\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$$ Recognizing that $\zeta(2)-\sum_{n=1}^k\frac1{n^2}$ can be written as $\psi_1(1+k)$ where $\psi_1(z)$ is the trigamma ...
4
votes
1answer
109 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 it,...