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.
1
vote
1answer
54 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
votes
3answers
194 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 ...
3
votes
2answers
54 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 ...
1
vote
1answer
58 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.
6
votes
1answer
156 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} + ...
0
votes
1answer
60 views
Reformulation of riemann zeta
Does this extend to $\mathbb{C}$?
$\displaystyle ζ(x) = \int_0^{\infty} \frac{ 1}{\lfloor t\rfloor ^x} dt$, where for $0 \leq t < 1$ we say that $\lfloor t \rfloor = 1$.
0
votes
0answers
40 views
Prove that $f(s)=εf(2-s)$
Let $f$ be an analytic function defined by
$$f(s)=N^{-s/2}(2π)^{s}Γ⁻¹(s)∑_{n=1}^{∞}(a_{n}/n^{s})(F_{n}(s-1)-εF_{n}(1-s))$$
where $$F_{n}(t)=Γ(t+1,2πn/√N).(√N/2πn))^{t+1},ε=±1$$ and ...
3
votes
0answers
71 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 ...
5
votes
1answer
101 views
Why $p$-adically interpolate?
I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
5
votes
1answer
91 views
Serre's proof that zeta function is meromorphic
I try to understand the proof of Chap. VI, n° 3.1, Prop. 10 in Serre's "A course in arithmetic" (page 70). The goal is to prove that zeta-function can be written as
\begin{align*}
...
3
votes
1answer
225 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 $$\pi \cot(\pi x)=\sum_{k \to \infty}^{\infty} ...
1
vote
1answer
45 views
Computing single summands of a zeta function
Given a zeta function
$$\zeta(s)=\sum_{n=1}^\infty |\lambda_n |^{-s},$$
I can do many tricks to get certain information. For example $\zeta'(0)$ might relate to the determinant of the operator where ...
0
votes
0answers
68 views
Ramanujan's claim [duplicate]
I was reading about the mathematician Ramanujan, and he claimed that 1+2+3+4+5+...=-1/12, and that this is related to the Riemann Zeta Function. Can someone explain the relationship? (By the way, I ...
2
votes
1answer
127 views
Convergence of the Fourier Transform of the Prime $\zeta$ Functions
I think I found a way to write the truncated Prime $\zeta$ function like this:
$$
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}}
...
3
votes
0answers
48 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 $\eta$:
$$ \eta(s) = 2^{s-1} \sum_{c=0}^\infty \left( ...
1
vote
2answers
75 views
Estimating the integrated Tchebychev function and calculating its error
I would like to understand how to derive (2) from (1) below.
Problem:
If $\psi_1$ is the integrated Tchebychev function below
$$\psi_1(x)=\frac{1}{2\pi i} ...
10
votes
3answers
141 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
0answers
83 views
A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )
Consider $t_n$ as the Thue-Morse sequence.
Let $m$ be a positive integer and $s$ a complex number.
Odiuos Number
Now consider the sequence of functions below
$f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$
...
0
votes
0answers
28 views
Complex exponentials, Polylogarithms
How can I rewrite $\text{Li}{s}(e^{2\pi i\frac{a}{b}})$, in terms of hurwitz zeta function, I know I have to break it up into congruence classes and sum each one independtly but not sure how I would ...
7
votes
2answers
86 views
Dirichlet Series for $\#\mathrm{groups}(n)$
What is known about the Dirichlet series given by
$$\zeta(s)=\sum_{n=1}^{\infty} \frac{\#\mathrm{groups}(n)}{n^{s}}$$
where $\#\mathrm{groups}(n)$ is the number of isomorphism classes of finite ...
1
vote
0answers
50 views
Pre-requisites for studying zeta functions
I want to start reading about zeta functions on my own, so i want to know what are the pre requisites that i need?
P.S : I am an EE engineer and have done the basic college level mathematical ...
1
vote
1answer
123 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 ...
1
vote
0answers
37 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
30 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 ...
0
votes
0answers
60 views
Polya's fake Zeta function $ \zeta ^{*} (1/2+iz) $ and differential operator
i have heard that the Polya's fake function's zeros $ \zeta ^{*} (1/2+iz)=0 $ are realted to the Eigenvalues of the operator
$$ - \frac{d^{2}}{dx^{2}} + e^{2x} $$
with boundary conditions $ ...
1
vote
0answers
98 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 :
$$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...
2
votes
0answers
52 views
Gram's series for integral equation
The prime counting function $ \pi(x) $ satisfies the integral equation
$$ \log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$
and it has the solution in terms of Gram's ...
2
votes
1answer
63 views
Derivative of the Selberg $\zeta$-function
I want to compute the derivative of the Selberg $\zeta$-function:
$$ \mathcal{Z}(s)=\prod_{\gamma \; \text{primitive}} \prod_{n=0}^\infty (1-e^{-l(\gamma)(n+s)}); \qquad \Re(s)>1.$$
Where ...
6
votes
1answer
133 views
Zeta functions of groups and an identity of Ramanujan
Zeta functions are being developed for all sorts of mathematical objects these days. One general situation is that of zeta functions of groups. If $G$ is a finitely generated group then we let
...
16
votes
3answers
607 views
A Geometric Proof of $\zeta(2)=\frac{\pi^2}6$? (and other integer inputs for the Zeta)
Is there a known geometric proof for this famous problem? $$\zeta(2)=\sum_{n=1}^\infty n^{-2}=\frac16\pi^2$$
Moreover we can consider possibilities of geometric proofs of the following identity for ...
5
votes
1answer
143 views
Zeta-like function with offset
Is there a known function of the form:
$$\zeta(s,a) = \displaystyle\sum_{n=1}^\infty\frac{1}{n^s+a},$$
and if so what is its name?
2
votes
3answers
194 views
Riemann Zeta formula
can anyone check if this formula is plausible ??
$$ \frac{1}{\zeta (s)} = \sum_{n=0}^{\infty}\frac{ (-\pi)^{n}(s-1)s}{2n!(s+2n)(s+2n+1)} $$
according to the authors this formula would be valid only ...
5
votes
1answer
116 views
On the Dirichlet beta function sum $\sum_{k=2}^\infty\Big[1-\beta(k) \Big]$
Given the Dirichlet beta function,
$$\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] = ...
2
votes
1answer
255 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 ...
9
votes
2answers
460 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 ...
3
votes
0answers
127 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 ...
3
votes
3answers
185 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 ...
11
votes
1answer
228 views
What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?
We have the following evaluations:
$$\begin{aligned}
&\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\
&\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
1
vote
2answers
109 views
Is there a Riemann hypothesis for the Hasse-Weil zeta function, generally?
What form does the Riemann hypothesis have for a global L-function?
8
votes
1answer
178 views
What is the $p$-adic zeta function?
I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in ...
5
votes
0answers
119 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 ...
0
votes
1answer
175 views
What is the fractional derivative of the function $\pi \cot (\pi x)$?
What is the fractional derivative of the function $\pi \cot (\pi x)$?
I derived the following expression:
$(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\gamma ) \zeta (p+1,q)}{\Gamma ...
1
vote
1answer
102 views
Number of projective points on a curve
This is a continuation of this previous problem I asked about.
From Ireland and Rosen's Number theory book(ch.11 ex#12)
EDITED: Let $C_{1}$ be the curve $y^2=x^{3}-Dx$ over $\mathbb{F}_{p}$, where ...
3
votes
1answer
204 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 ...
2
votes
0answers
102 views
Determinant of the Laplacian of a surface is this correct?
given a surface with metric $ g_{ab} $ i would like to evaluate the functional determinant of the Laplacian in the form
$ - \partial _{s} \zeta (0,E^{2})=\log\det( \Delta + E^{2}) $
then i need to ...
2
votes
0answers
85 views
Constant term of zeta binomials
Let's have the following zeta binomial $\sum\limits_{n=1}^\infty (1/n-1/(n+1))^k$, where $k$ a natural number and $k>1$. From the expansion of these binomials we obtain polynomials of $\pi$ where ...
1
vote
2answers
137 views
Series involving zeta functions
Let's have the following zeta function binomial: $\sum\limits_{n=1}^\infty \left(\frac{1}{n}-\frac{1}{n+1}\right)^2=\pi^2/3-3$.
Does anyone know the limit of the following zeta function binomial ...
1
vote
0answers
27 views
Number of energies of the free Laplacian.
Given the Selberg trace formula, and the fact that the eigenvalues of the operator $\Delta -1/4 =T$ are the zeros of the Selberg zeta function, then would it be correct to say the number of ...
4
votes
1answer
137 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
...
7
votes
2answers
566 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 ...
