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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11
votes
3answers
2k views

Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$ I found this result very ...
57
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( ...
10
votes
2answers
510 views

How to find $\zeta(0)=\frac{-1}{2}$ by definition?

I would like to know how we can find the following result: $$\zeta(0)=-\frac12$$ Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$ to find this?
23
votes
3answers
467 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?
8
votes
2answers
648 views

Approximation of Products of Truncated Prime $\zeta$ Functions

The problem arose, while I was looking at products of power prime zeta functions $$ P_x(ks)=\sum_{p\,\in\mathrm{\,primes}\leq x} p^{-ks}, $$ with $k\in \mathbb{N}$ and $s=it$ with real $t$. By using ...
3
votes
0answers
371 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} ...
33
votes
5answers
2k views

Does $\zeta(3)$ have a connection with $\pi$?

The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Several ...
15
votes
5answers
277 views

Evaluating $\sum_{k=1}^{\infty}\frac{\sin\left(k\theta\right)}{k^{2u+1}}$ with multiple integrals

I am trying to evaluate $$\sum_{k=1}^{\infty}\frac{\sin\left(k\theta\right)}{k^{2u+1}}\tag{$u\in\mathbb{N}$}$$ using some results I've got. I know that ...
2
votes
1answer
195 views

Convergence of the Fourier Transform of the Prime $\zeta$ Functions

I think I found a way to write the truncated Prime $\zeta$ function like this: $$ P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} ...
5
votes
1answer
331 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 ...
7
votes
3answers
410 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 ...
8
votes
1answer
543 views

Apéry's constant ($\zeta(3)$) value

I tried to find some proofs about the Apéry's constant, but I didn't find any intuitive proof. Is this constant given by the "brutal force" summing of $1 + \frac{1}{2^3} + \frac{1}{3^3} + ...
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
301 views

Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?

Motivation: I was looking at the approximation of the truncated Prime $\zeta$ function $$ P_x(s)=\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O \left(\cdot \right) $$ (to be found here with or ...
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, ...
23
votes
0answers
687 views
+100

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) + ...
10
votes
2answers
237 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 ...
11
votes
2answers
617 views

Why do mathematicians care so much about zeta functions?

Why is it that so many people care so much about zeta functions? Why do people write books and books specifically about the theory of Riemann Zeta functions? What is its purpose? Is it just to ...
7
votes
1answer
96 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 \\ ...
5
votes
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 ...
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 ...
12
votes
1answer
187 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
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 ...
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
votes
0answers
52 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: ...
1
vote
0answers
83 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 ...
1
vote
0answers
129 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 ...
1
vote
1answer
58 views

What's the probability to find a value of $t<T$ where $|P_k(it)|<\epsilon$?

Given $P_k$, the truncated Prime $\zeta$ function, defined like $$ P_k(it):=\sum_{n=1}^k p_n^{it}, $$ where $p_n$ is the $n$th prime. What's the probability to find a value or range of $t$ less than ...
0
votes
1answer
79 views

How to solve this summation (Lerch Transcendent)?

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

Is the Mean Value of $|P_k(it)|$ equal to $\sqrt{k}$?

Let $P_k$ be the truncated Prime $\zeta$ function, like $$ P_k(it)=\sum_{n=1}^k p_n^{it}, $$ with $p_n$ being the $n$th prime. Numerics seem to indicate that the mean value of $|P_k|$ taken over all ...