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
votes
1answer
48 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 ...
3
votes
0answers
162 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 ...
3
votes
0answers
113 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
89 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 ...
7
votes
1answer
240 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 ...
17
votes
3answers
742 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
166 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
265 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 ...
6
votes
1answer
244 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
304 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 ...
11
votes
2answers
653 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
192 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
364 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 ...
15
votes
1answer
389 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} = ...
2
votes
1answer
261 views

Is there a Riemann hypothesis for the Hasse-Weil zeta function, generally? [duplicate]

What form does the Riemann hypothesis have for a global L-function?
10
votes
1answer
307 views

What are the branches of 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
151 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
249 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
114 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
316 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
119 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
92 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
169 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
32 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
178 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 ...
8
votes
2answers
655 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 ...
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} ...
11
votes
1answer
318 views

Regularizing divergent series and Bernoulli numbers

Here is the actual problem I need a proof for: (I made an error writing down the equation initially) $$B_{k+1} = \frac{(-1)^k}{2^{k+2}-2} \sum_{q=0}^{k} \binom{k+1}{q} 2^q B_q$$ Below is my ...
7
votes
1answer
454 views

Reference request: $L$-series and $\zeta$-functions

Does anyone know a good book, lecture note, article etc. on $L$-series (Dirichlet, Hecke, Artin) and $\zeta$-functions in number theory? I'm especially interested in material explaining the following: ...
10
votes
2answers
547 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?
8
votes
0answers
308 views

New generalization of Riemann Zeta?

I am interested in the following generalization of the Riemann Zeta function: $$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$ This is most closely related ...
5
votes
2answers
191 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 $ ...
2
votes
1answer
283 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 ...
3
votes
4answers
395 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, ...
4
votes
0answers
147 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 ...
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 ...
5
votes
1answer
329 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 $$\prod_{k=0}^{\infty}\left(1+x^{2^{k}}\right)$$ ...
2
votes
1answer
476 views

Is Riemann Zeta Function symmetrical about the real axis?

From wikipedia, http://en.wikipedia.org/wiki/Riemann_zeta_function "Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the ...
5
votes
1answer
358 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 ...
5
votes
0answers
138 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) ...
1
vote
2answers
144 views

On functions similar to Hurwitz zeta function

Denoted as $\zeta(s,a)$ for a > 0 Where do I find topics on the Hurwitz zeta function for a < 0? Any links or resources would be appreciated. (Please dont mention wiki or mathworld) Thanks
3
votes
0answers
257 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 ...
2
votes
2answers
442 views

Some basic questions about the Selberg zeta function

I'm trying to learn about the Selberg zeta function, but it seems like introductory texts assume more knowledge of Riemannian geometry than I'm comfortable with. I have some basic questions that ...
2
votes
2answers
190 views

Proving identity $\displaystyle\sum_{j\geq 1}[(j+t)^{-1}-j^{-1}]=\displaystyle\sum_{k\geq 1}\zeta (k+1)(-t)^{k}$

Motivation: In S.J. Patterson's An introduction to the theory of the Riemann Zeta-Function it is proved (p.132) that $\displaystyle -\Gamma ^{\prime }(t)/\Gamma (t)=\gamma +t^{-1}+\underset{j\geq ...