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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27 views

zeta function and probability divisible by k and choosing squares

I have some difficulty starting with this question: For $s>1$, set $\zeta(s)=\sum_{n=1}^\infty n^{-s}$, and define $p: \mathbb{N} \rightarrow [0,1]$ by $p(n)=\frac{n^{-s}}{\zeta(s)}$. One can ...
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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
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99 views

Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?

From this question: http://mathoverflow.net/questions/134600/zetax-in-terms-of-zetax-zeta1-x-gamma-psi it seems that Zeta can be expressed through its derivative: $$\zeta(1-x) = ...
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1answer
45 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 ...
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1answer
83 views

Difference between Meromorphic and Analytic Continuation

We have that $\frac{X}{spec \mathbb{Z}}$, a scheme of finite type. Consider $\zeta(X,s)= \prod_{x\in{X}} \frac{1}{(1-Nx^{-s}}$, with $Nx$ the norm. I didn't catch what my teacher was saying and he is ...
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1answer
122 views

proof regarding zeta function of a curve from Ireland and Rosen's “A Classical Introduction to Modern Number Theory”

In chapter $11$ section $5$ titled "the last entry" the authors state that the number of solutions to the congruence $x^2+y^2+x^2y^2 \equiv 1 \mod p$ is $p+1-2a$, where $p=a^2+b^2$ and $a+bi \equiv ...
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1answer
56 views

A limit involving the Hurwitz/generalized Riemann zeta function

I want to show that $$ \lim_{s \to 1} \Big( \zeta(s,a) - \frac{1}{s-1} \Big) = - \psi(a)$$ where $\zeta(s,a)$ is the Hurwitz zeta function and $\psi(a)$ is the digamma function. The only thing I ...
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1answer
107 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 $\lfloor t \rfloor = 1$.
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1answer
60 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 ...
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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 ...
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1answer
253 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 ...
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0answers
16 views

What is the Möbius analoge for Ihara's $\zeta$ function?

The Dirichlet series that generates the Möbius function is the (multiplicative) inverse of the Riemann zeta function; if s is a complex number with real part larger than 1 we have $$ ...
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35 views

Divergence Dedekind zeta function

Let $K$ be a number field, $\mathcal O_K$ be its ring of integers, T a positive integer and $N$ the norm function. Give an upper bound (in T) for $$\sum_{I\leq \mathcal O_K: N(I)\leq T} ...
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1answer
31 views

Convergence of the series for the Dedekind zeta function

The Dedekind zeta function of an algebraic number field $K$ is defined as $\zeta_K(s)\mathrel{\stackrel{\rm def}=} \sum\limits_{I \subset\mathcal O_K} \frac{1}{(N_{K/\mathbb Q}(I))^s}$, where ...
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26 views

Generalized Hurwitz Zeta function

Let $d\ge 1$ be an integer, $a>0$ be a real number and let $\vec{s} := (s_0,\cdots,s_{d-1})$ where all the components are strictly bigger than one. We generalize the zeta function to higher ...
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38 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 ...
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58 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} ...
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40 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 ...
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76 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 ...
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107 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 ...
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33 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?
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115 views

The alternating zeta function and functional equation

The Dirichlet eta function (the alternating zeta function) is given by $$η(s)=∑_{n=1}^{∞}(-1)ⁿ⁻¹/n^{s}$$ The functional equation for $η(s)$ is given by $$η(s)=ϕ(s)η(1-s)$$ where ...
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42 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
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26 views

What are Selberg class functions of degree two?

My question is: What are Selberg class functions of degree two. I know about the Selberg class. But I am not able to understand what means by degree two.
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31 views

How one can determine the rapidly convergente series?

The motivation to this question can be found in http://wstein.org/books/bsd/bsd.pdf In page 9 the author claimed that: 1.4.1. Approximating the Rank. Fix an elliptic curve $E$ over $Q$. The usual ...
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61 views

Prove that the only negative real zeroes are at the integers

Let $$L(C,s)=\prod_{p\mid\Delta}(1-a_{p}p^{-s})^{-1}\cdot\prod_{p\nmid\Delta}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse-Weil $L$-function of ...
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86 views

Conjectures about zeta functions and poles

Let $p^*_n$ be the $n$ th element of a subset of primes such that $p^*_{n+1}>p^*_n$ and $p^*_n < O((n+2) ln((n+2))^3)$. Define $f(z)$ as the analytic continuation of $\prod_{n>0} ...
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143 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 ...
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61 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 ...
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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 ...
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1answer
50 views

Formula for tangent derivatives, how to prove?

How to prove? $$(\tan x)^{(s-1)}=\pi^{-s}\Gamma(s)\left(\zeta\left(s, \frac12-\frac x\pi\right)+(-1)^s\zeta\left(s, \frac12+\frac x\pi\right)\right) $$
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Does this summation (involving binomial) have a closed form? If so, what is it?

The following sums are the ones I'm interested in: $\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}$ $\lim_{\Omega\rightarrow\infty}{\sum_{i=m}^{\Omega}{\binom{i}{m}i^{-k}}}$ I already know that ...
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1answer
22 views

Generating function for the zeta function

Given a Hamiltonian $H$, with a spectrum of eigenvalues $\lambda$, you can define its zeta function as $\zeta_H(s) = tr \frac{1}{H^s} = \sum_{\lambda}^{} \frac{1}{\lambda^s}$. Subsequently, the log ...
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1answer
83 views

How to solve this summation (Lerch Transcendent)?

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

Infinite sum and equality between coefficients of the same index

I have two infinite sums that forms an equality: $$\sum_{n=1}^{\infty} \left(\zeta(2n)\frac{x^{2n}}{\pi^{2n}}\right) = \sum_{n=1}^\infty ...
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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 ...
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36 views

Provide me notes on Riemann zeta function to boast my knowledge to use in Research on Analytical Number Theory

I need your help. I want to study the Riemann zeta function from the very basic level, its concepts, theorems, solved problems etc. I am assigned one problem from Analytical Number Theory related to ...
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24 views

Kronecker Zeta function

If we define the Kronecker symbol K(a,n) as at Wikipedia, can we define $$\zeta_K(s,a) = \sum_n \dfrac{K(a,n)}{n^s}$$? If so, what does it equal?
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17 views

generating function for the Hurwitz Zeta

we know from the digamma function $$ \Psi (z+1)= -\gamma -\sum_{k=1}^{\infty}\zeta(k+1)(-z)^{k} $$ my question is if there is a similar formula for $$ f(a)+ ...
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zeta-function regularization $\prod_{m,n\in \mathbf{Z}}(m+n\tau+u)=iy^{\frac{1}{2}} q^{\frac{1}{12}} \prod_{n=1}^{\infty} (1-yq^n)(1-y^{-1}q^{n-1})$

I'd like to prove(explicilty compute) the following relation \begin{align} \prod_{m,n\in \mathbf{Z}}(m+n\tau+u)=iy^{\frac{1}{2}} q^{\frac{1}{12}} \prod_{n=1}^{\infty} (1-yq^n)(1-y^{-1}q^{n-1}) ...
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36 views

Generalize a trick with Dirichlet series to algebraic number theory

I am not able to generalize the following equality involving Dirichlet series : ...
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28 views

Can you provide a lower bound on $|\zeta(\sigma+it)|$ for $\sigma$<0?

To prove the theorem 9.12 of "The theory of the Riemann Zeta-Function", I have to show that $|\zeta(\sigma+it)|$ > $A|t|^c$ for $\sigma$<0 and sufficiently large t. How can I show this?
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23 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 ...
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51 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 ...
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24 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 $\tan x$. ...
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64 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.
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35 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: ...
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33 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 !
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81 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 ...