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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1answer
41 views

Steps inbetween? Weil's zeta function

Why is it that, that is "just" what the Zeta function is? What happened in between? I messed around with it for roughly an hour and couldn't get it to come out right. The second photo is just for ...
-2
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0answers
62 views

It is possible to use the Zeta Function as primality test? [closed]

It is possible to use the Zeta Function as primality test? $$\displaystyle\sum_{n=1}^\infty\dfrac1{n^s} = 1+\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+ ... $$ Where can I find the non-trivial zeros ...
3
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0answers
37 views

What is the link between Primes and zeroes of Riemann zeta function?

Usually Riemann hypothesis is introduced along this lines. (1.1) Geometric progressions were known since forever (1.2) Euler factorization links a product of primes and a sum of natural numbers (1.3) ...
10
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3answers
59 views

Calculate the number of integers in a given interval that are coprime to a given integer

We can calculate the number of integers between $1$ and a given integer n that are relatively prime to n, using Euler function: Let $p_1^{\varepsilon1}\cdot p_2^{\varepsilon2} \cdots p_k^{\varepsilon ...
0
votes
1answer
74 views

How was the integral for Zeta Function created

How was the zeta function integrated from $$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^{s}}$$ To $$\zeta(s) = \frac{1}{\Gamma (s)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx$$ I've tried googling ...
1
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0answers
34 views

Is there any product formula for local zeta function?

Suppose that $V$ is a non-singular $n$-dimensional projective algebraic variety over the field $\mathbb{F}_q$ with $q$ elements. The local zeta function $Z(V, s)$ of $V$ (sometimes called the ...
0
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1answer
27 views

Expressing Tornheim sums in terms of Riemann's Zeta

If $$T(a,b,c)=\sum_{r\geq1}\sum_{s\geq1} \frac{1}{r^as^b(r+s)^c}$$ How to prove that : $$T(3,1,2)=-\frac13 \zeta(6)+\frac{\zeta^2(3)}{2}$$ I tried some algebraic manipulations but did not work. Can ...
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0answers
52 views

Geometric and arithmetic Frobenius

I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the geometric Frobenius $F: X \mapsto X$ is defined by fixing every element of the ...
0
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0answers
34 views

Analytic nature of Reimann Zeta function at Re(s)=1 and about the domain of holomorphicity of Zeta function

How can we prove that $\zeta(1+it)$ is analytic for $t \neq 0$ without using the functional equation? And before doing the analytic continuation of $\zeta(s)$, is it possible for the zeta function to ...
3
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0answers
40 views

Good approximation to zeta function in the critical strip by smoothed sum

I'm self-studying analytic number theory from terry tao's blog, there is an exercise (Exercise 33) from the blog that I cannot solve: Let ${\eta: {\bf R} \rightarrow {\bf C}}$ be a smooth ...
3
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3answers
460 views

Interesting Series with Zeta Function

I was trying to find another representation for the value of an integral when I found the following series: $$f (z)=\sum_{n \in \Bbb N} (-z)^{n-1}\frac {(2^n-1)}{2^n}\zeta (n+1) $$ For $|z|<1$ and ...
3
votes
0answers
26 views

Sum of integers and zêta functions

I am working on generalizing some works from the usual rational case to general number fields. That implies some technical changes I am not really at ease with. For instance: $$\sum_{m \leqslant X} m ...
1
vote
0answers
32 views

Estimate of the log derivative of zeta function in the classic zero-free region

We know that the Riemann zeta function $\zeta$ has no zeros in the region $\{\beta+it:\beta>1-\frac{c}{\log(2+|t|)}\}$, where $c>0$ is an absolute constant. This is known as the classical zero-...
1
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0answers
42 views

Proving $\zeta(3)\in\mathbb{R}\setminus\mathbb{Q}$ using modular forms

I am looking for references proving $\zeta(3)\in\mathbb{R}\setminus\mathbb{Q}$ using modular forms, like this paper written by F. Beukers. Does anybody know some different papers or books? Thanks.
0
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0answers
30 views

Weil conjectures - If two varieties have the same of Fq^d - valued points for all d >> 0, then they have the same Hasse - Weil function

I was working on the following exercise for fun, and I haven't really gotten anywhere with it. Let Z( X; t) be defined as exp ( $\sum_{r= 1}^{\infty} N_r t^r/r$), where $N_r$ is the size of X($\...
3
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2answers
89 views

Show me how to evaluate $\int_0^1\int_0^1\frac{[-\ln(x)]^s}{1-xy}dxdy=\frac{\zeta(s+2)}{\Gamma(s+2)}$

Double integrals (1) $$\int_0^1\int_0^1\frac{[-\ln(x)]^s}{1-xy}dxdy=\frac{\zeta(s+2)}{\Gamma(s+2)}$$ (2) $$\int_0^1\int_0^1\frac{[-\ln(xy)]^s}{1-xy}dxdy=\zeta(s+2)\Gamma(s+2)$$ Where $\sum_{n=0}^{\...
3
votes
1answer
67 views

Show that $\zeta'(0)=-\frac{1}{2}\ln(2\pi)$

I started with the functional equation which was derived in class, $$ \zeta(s)=2^s\pi^{s-1}\sin(\frac{\pi s}{2})\Gamma(1-s)\zeta(1-s) $$ and took the logarithmic derivative of both sides to get $$ \...
1
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0answers
18 views

Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=$ for Re$(z)>0$. [duplicate]

Show that $(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=\frac{1}{\Gamma(z)}\int\limits_{0}^\infty \frac{t^{z-1}}{e^t+1}dt$ for Re$(z)>0$. Not sure how to get started on this, we ...
2
votes
1answer
36 views

What is the relationship between all the dynamical zeta functions and the number theoretical zeta functions?

One can associate to any dynamical system a zeta function based on counting the number of fixed points of the iterates of the transformation. Explicitly we have: $$\zeta_{A} = exp \left( \sum_{n=1} \...
4
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1answer
46 views

Prove that this summation evaluates out to $\zeta(2)-1$

I am aware of the following identity: $$\sum_{m=1}^\infty \left(\frac{1}{m}-\left(\zeta(2)-\sum_{n=1}^m \frac{1}{n^2}\right)\right)=\zeta(2)-1$$ I can't quite figure out how to prove this result. ...
4
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2answers
119 views

Find: $\zeta \left( 3,1,1,1 \right)$

While solving a summation, I came across this: $$\zeta \left( 3,1,1,1 \right)=? $$ I'm new to multiple zeta values. That's why I couldn't find this. So my question is does a closed form exist ...
5
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2answers
110 views

Generalizing $f(n)=\int_0^\infty \frac{1}{e^{x^n}+1}=\left(1-2^{(n-1)/n}\right )\zeta(n^{-1})\Gamma(1+n^{-1})$

I have come up with the following solution to this integral, but is just incomplete to my standards $$f(n)=\int_0^\infty \frac{1}{e^{x^n}+1}=\left(1-2^{(n-1)/n}\right )\zeta(n^{-1})\Gamma(1+n^{-1})$$ ...
4
votes
3answers
90 views

Asymptotic formula for $\prod_{k=1}^{\infty}\zeta (2kn)$

Suppose $n\geq 1$ is a positive integer. Can we find an asymptotic formula for this product below. $$\prod_{k=1}^{\infty}\zeta (2kn)=\zeta (2n)\zeta (4n)\zeta (6n) \cdots$$ I tried to use $\zeta (2n)...
9
votes
1answer
108 views

Convergence of zeta functions for schemes of finite type over the integers

In his lecture "Zeta functions and $L$-functions", Serre presents a very elegant proof of the convergence of the zeta function $ \zeta (X,s) = \prod_{x \in |X|} (1- N(x)^{-s})^{-1}$ in the half plane ...
4
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4answers
168 views

How to show $\zeta (1+\frac{1}{n})\sim n$

How to show $\zeta (1+\frac{1}{n})\sim n$ as $n\rightarrow \infty$ where $\zeta$ is the Riemann zeta function. And can we say $\lceil \zeta (1+\frac{1}{n}) \rceil=n$ for any positive integer $n\geq 1$...
2
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0answers
30 views

Comparing Dedekind zeta functions

It is known that non-isomorphic number fields can share the same Dedekind zeta function. However, there don't appear to be any examples of very low degree so in these cases the zeta function must ...
1
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2answers
71 views

Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$

$$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$ for $t\in\Bbb{N}$ How might one represent $a,b$ in terms of $t$? (Note that $a,b\in \Bbb{N}$) If possible, could one also ...
6
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0answers
56 views

Why was the zeta function introduced?

I know the 'Zeta Function' is very useful in Mathematics, and that it has relations with many other functions (such as the 'Gamma Function'). I also know the 'Zeta Function' $\zeta(s)$ is defined as: ...
1
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0answers
47 views

Euler Product Formula - Zeta Function

For $s \in \mathbb{C}$ and $\sigma = \Re(s)>1$, $$\zeta(s) = \prod\limits_{p \in \mathbb{P}}\left(1 - \frac{1}{p^s}\right)^{-1}$$ My question is: is the above correct? Or should the $s$ be ...
2
votes
2answers
139 views

Assigning values to a divergent integral?

Question If I can assign the series of the zeta function to: $$ \zeta(-1) \to 1+2+3+\dots$$ why can't we assign the integral $$ \int_{0}^{\infty} x dx \to 0$$ and it still have some physical ...
2
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0answers
34 views

Can someone expain to me what's going on (binomial coefficient)?

I'm watching this proof for $\zeta(2n)$ on YouTube. This is what I can understand so far: $${s\over e^{s} -1} = \sum^{\infty}_{n=0} {\beta_n\over n!} s^n$$ Where $\sum^{\infty}_{n=0} {\beta_n\over ...
7
votes
1answer
209 views

Evaluate $\displaystyle \sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x.$

I have some trouble in evaluating this series $$\sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x$$ I tried to calculate the integral first, but after that I found the ...
1
vote
1answer
21 views

Bounds for zeta function residue

Let $K$ be an algebraic number field and let $c = c(K)$ denote the residue at $s = 1$ of its zeta function. It is known Wikipedia: class number formula that c can be determined via $$c = \frac{2^r (2\...
2
votes
2answers
179 views

Where do these infinites Tao is talking about come from?

I was reading about how 1+2+3+4... !=-1/12 (which is something that drove me crazy when I first heard about it in a Numberphile video) in an article by Terence Tao. He says that -1/12 is in fact -1/12 ...
3
votes
2answers
68 views

Convergence of prime zeta function for $\mathfrak R(s)=1$?

By doing some estimates for the partial sums of the Prime zeta function $P(s)=\sum_p p^{-s}$ for $\mathfrak R(s)=1$ I got that $P(1+i\alpha)$ converges for every $\alpha\neq0$... Since I did not ...
0
votes
1answer
13 views

What is the meaning of cyclically equivalence classes of multiple indices?

Can you please give me an example for the cyclic equivalence classes of multiple indices on the following set? $$ I(w,d)=\{\ (k_1,...k_d)\mid k_1+\ldots+k_d=w, k_1,...k_d \ge 1\,\}$$ where $ w$: ...
1
vote
1answer
31 views

Seeking possibility of more elementary means of evaluating an improper integral.

It can be shown that $\int_0^\infty -\log{(1-e^{-x})}=\zeta(2)$ by expanding out the integral as $\log(1-z)$, exchanging summation and integration, then summing up the integrals. I am wondering if ...
3
votes
1answer
68 views

Is there a closed form for the product of odd zetas?

$$\prod_{n=1}^\infty \zeta(2n+1)=\zeta(3)\zeta(5)\cdots$$ I have only managed to prove that this converges due to comparison with Euler's formula for $\zeta(2n)$ Is there a closed form for that ...
0
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1answer
51 views

How do I evaluate this sum :$ \sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}\log n}{n^s}$?

How do I evaluate this sum :$$ \sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}\log n}{n^s}$$ Note : In wolfram alpha it is convergent for $Re(s)>1$ .!! Thank you for any help
1
vote
1answer
285 views

All Ihara $\zeta$ functions for planar $k$-regular graphs with a given set of faces are equivalent

This sounds like a simple piece of math (which got a long story over time, thanks for reading!) and the consequence seems surprising. At least to me. Here it is: It boils down to comparing two ...
0
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1answer
41 views

Books on Zeta Regularization Product

Does anybody know some book on zeta regularization, and the zeta regularization product? I'm quite interested on the topic but I would need a book with some review...
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0answers
201 views

A map from zeros of $\zeta(s)$ to zeros of $C(s)?$

Let $P(s),C(s),\zeta(s)$ be the prime zeta function, the analogous composite zeta function, and the classical zeta function. I do not know whether it is known that there are infinitely many zeros of ...
1
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1answer
36 views

Does the Weil conjecture work with 0 dimensional varieties?

Suppose I have some inseparable quadratic polynomial $p(x)$ over $\mathbb{F}_q$ which has a pair of roots in $\mathbb{F}_{q^2}$. If I compute the zeta function of the variety cut out by $p(x)$ (in ...
1
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0answers
27 views

Proof that $\zeta_P$ never does the following…

Let's assume that $\zeta_P(s)$ is the prime-zeta function or: $$\zeta_P(s)=\sum_{n\in P} \frac{1}{n^s}$$ I noted that if $\forall s\in \Bbb{Q},s\not =0, \zeta_P (s)\not\in \Bbb{Q}$ I cant really ...
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0answers
39 views

Zeta function of $y^2 = x^3 - x$ over Fp

Zeta function of $y^2 = x^3 - x$ over Fp, where p = 3(mod 4) Can someone give an explanation of a zeta function? I've tried researching it, and I cannot seem to understand. Is there some kind of ...
0
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0answers
7 views

does the $\zeta_K$ function of a function field determine the genus of that function field?

Let $K_1$ and $K_2$ both be function fields over a finite field (or algebraic curves, if you like) with zeta functions $\zeta_{K_1}$ and $\zeta_{K_2}$. Say that $\zeta_{K_1} = \zeta_{K_2}$ - so that ...
4
votes
0answers
125 views

Finding zeta function of an elliptic curve

Let p=3 (mod 4) be a prime, and $E/F_{p^r}$ be the elliptic curve given by $y^2 = x^3 − x$ Find the zeta-function of $E/F_p$ and use it to determine $|E(F_{p^r} )|$ for all r>0.
3
votes
2answers
59 views

Closed form of the series $\sum\frac{\ln(n)}{n^a}$

I want to know if there exists, and how to arrive at, a closed form of this infinite sum: $$S_a=\sum_{n=1}^\infty \frac{\ln(n)}{n^a}$$ I know the series converges at least for every $a>1$ by the ...
3
votes
0answers
116 views

Zeros of the prime zeta function

A basic confusion about zeros of the prime zeta function $P(s).$ Let $s= \sigma+i~t$ with $\sigma>0.$ Letting $C(s)$ be the corresponding composite zeta function we have $$(1)\hspace{15mm}P(s)+C(...
5
votes
1answer
108 views

Mysterious Inverse Mellin transform using residue theorem

The origin of this problem lies in the explanation of the evaluation of the series $\sum_{n\geq1}\frac{\cos(nx)}{n^2}=\frac{x^2}{4}-\frac{2\pi}{4}+\frac{\pi^2}{6}$ see this link ( Series $\sum_{n=1}^...