Tagged Questions

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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0
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0answers
14 views

convergence of 2 series in the critical strip

let us define 2 series: $$A=\sum_{k=1}^{+\infty}(-1)^{(2k+1)}\frac{\ln(2k+1)}{(2k+1)^s}$$ $$B=\sum_{k=1}^{+\infty}\frac{\ln(2k)}{(2k)^s}$$ Define $$ s=\alpha + \beta i$$ Does $\frac{A}{B}$ go to ...
7
votes
1answer
71 views

Integral involving hyperfactorial

I'm trying to prove that: $$ \int_0^1 \ln\left(K(x)\right)\space dx =-\zeta'(-1)=\ln(A)-\frac{1}{12} $$ Where $A$ is Glaisher Kinkelin's constant and $K(x)$ is a generalization of the hyperfactorial ...
7
votes
0answers
53 views

Question on the paper Donal F. Connon, “Some integrals involving the Stieltjes constants”

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
3
votes
1answer
39 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 ...
0
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0answers
15 views

Zeta function and heat kernel

It is easy to prove that zeta function $$\zeta_{\Lambda}(s)=\sum \frac{1}{\lambda_{n}^{s}}$$ and trace of heat kernel $$K_{\Lambda}(t)=\sum e^{-\lambda_{n}t}$$ satisfy the relashion ...
0
votes
0answers
26 views

Singularities of zeta function

I have to prove (if $\gamma \ne 0$) that there is a analytic continuation for $\Re s >0$ of the function $$f(s)=\frac{\zeta (s)^2 \zeta(s-i\gamma )\zeta(s+i\gamma ) }{\zeta(2s)} $$ and that this ...
0
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0answers
10 views

What is the closed form for generation function of $\xi(2x)$ (Riemann Xi)?

I wonder whether the following coincidence is just random. Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. ...
1
vote
0answers
20 views

On the partial zeta function

Let $F$ be a number field and $S$ be a finite set of places of $F$ including archimedian places. Let $\zeta^S(s)$ be the partial L-function, that is the meromorphic continuation of the product of ...
1
vote
0answers
32 views

Double series of Harmonic Numbers

In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} ...
4
votes
1answer
86 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 ...
1
vote
1answer
44 views

Pulsating waves of zeta function

Below is an animation of the partial sums of $\operatorname{li}x-2\Re\sum_{k=1}^{N}\operatorname{Ei}(\rho_k \log x)-\log2+\int_{x}^{\infty}\dfrac{\text{d}t}{t(t^2-1)\log t},$ for $1\leq N\leq100$ ...
2
votes
1answer
40 views

How to solve the following delay differential equation?

What is the solution for the following equation? $$\frac\partial{\partial q}f(s,q)= \frac s2 f(s+2,q)$$ Note, it is known that the solution for $$\frac\partial{\partial q}f(s,q)= s f(s+1,q)$$ ...
7
votes
0answers
132 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
0
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0answers
49 views

Analytic Continuation of the zeta function

Is the analytic continuation of the Riemann zeta function to the upper half plane unique? I don't know much complex analysis, so I can't see why that is the case.
2
votes
0answers
53 views

Zeta zeros & Exponential Integral

From the oscillating part of an explicit formula for primes: $$\text{Re}(\operatorname{li}(x^\rho))\approx x\ \text{Re}(\operatorname{li}(x^{-\rho}))$$ $\rho_n$ may be replaced by any complex ...
2
votes
3answers
104 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 ...
4
votes
2answers
115 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 $ ...
1
vote
0answers
23 views

Zeta Function on a Finite Field - Koblitz

I am reading Koblitz p-adic analysis book and I am on page 111. The lemma is that $\zeta_{H_f}(T)$ has coefficients in $\mathbb{Z}$. I could follow the rest of the book from page 1 just fine until the ...
0
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0answers
32 views

Do all complex zeros of $Li_s(z)\,- \, Li_{1-s}(z)$ get the shape $s=\dfrac12 + \dfrac{k \, \pi }{\,\ln(2)}\,i$ when $z \rightarrow 0^{-}$?

This is a subquestion of this question on MO. Numerical evidence strongly suggests that when $z \rightarrow 0^{-}$ the complex zeros that lie in the critical strip $0 \lt \Re(s) < 1$ of: ...
2
votes
0answers
50 views

Intuitive explanation for $\zeta (2)=\frac{\pi^2}{6}$ [duplicate]

Using $f(x)=x^2$ Fourier' series, the proof for $\zeta (2)=\frac{\pi^2}{6}$ is pretty straight forward. I'm wondering if there is a more intuitive explanation for the equality, one that a layman could ...
2
votes
1answer
51 views

Proof or source for this Hurwitz Zeta function identity?

I need a proof or source for this identity: $ \zeta '\left(z,\frac{q}{2}\right)-2^z \zeta '(z,q)+\zeta '\left(z,\frac{q+1}{2}\right)=\zeta(z,q)2^{z}\ln 2$ Here the derivative means the derivative by ...
8
votes
3answers
321 views

Sum related to zeta function

I was trying to evaluate the following sum: $$\sum_{k=0}^{\infty} \frac{1}{(3k+1)^3}$$ W|A gives a nice closed form but I have zero idea about the steps involved to evaluate the sum. How to approach ...
4
votes
1answer
85 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 $$ ...
0
votes
0answers
29 views

A functional equation for a Dirichlet serie

I'm looking for a functional equation for the following Dirichlet serie $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ? Thank you !
1
vote
0answers
56 views

Plotting the pair correlation function for the zeta zeros /GUE

I am making a shameless request for instructions on how to plot this: from this page. I can see from here that normalizing the zeros is given by ...
8
votes
2answers
193 views

What is the sum of all complex integers?

In line with $$\zeta(-1)=-1/12$$ Could we, by considering $$f(s)=\sum_{a,b\in\mathbb Z,\;(a,b)\neq(0,0)}\frac{1}{(a+bi)^{s}}$$ Evaluate the sum of all complex integers?
5
votes
1answer
80 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) - ...
16
votes
2answers
447 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
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0answers
128 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
71 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
94 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
23 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
96 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
14 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
95 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
15 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
42 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
87 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
210 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
134 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
77 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
16 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
38 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
167 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
57 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 ...
-3
votes
2answers
134 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
75 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 ...
3
votes
0answers
47 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: ...