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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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 ...
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38 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.
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25 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 ...
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2answers
77 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 ...
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3answers
841 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 ...
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1answer
59 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 $$ ...
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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 ...
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4answers
5k 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 ...
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42 views

A functional equation for a Dirichlet series

I'm looking for a functional equation for the following Dirichlet series $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ?
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1answer
35 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} ...
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1answer
106 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 ...
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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 $$f(n)=\int_0^\infty \frac{1}{e^{x^n}+1}=\left(1-2^{(n-1)/n}\right )\zeta(n^{-1})\Gamma(1+n^{-1})$$ ...
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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. ...
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2answers
110 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 ...
4
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1answer
191 views

Infinite Series $\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$ [duplicate]

Evaluate: $$\sum_{k=1}^\infty\left(\zeta(2)-\sum_{n=1}^k\frac1{n^2}\right)^2$$ Recognizing that $\zeta(2)-\sum_{n=1}^k\frac1{n^2}$ can be written as $\psi_1(1+k)$ where $\psi_1(z)$ is the trigamma ...
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2answers
135 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 ...
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2answers
585 views

Simpler zeta zeros

Is it true that $$\lim_{y\rightarrow\infty}\dfrac{\sum_{n=1}^{y}n^{-1/2-iy}}{\zeta(1/2+iy)}=1$$ ? Below is a plot of $$\sum_{n=1}^{y}\dfrac{1}{n^{s}}\text{for }s=\dfrac{1}{2}+iy$$ set against its ...
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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 ...
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3answers
742 views

Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$

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?
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3answers
88 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 ...
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4answers
163 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 ...
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0answers
27 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 ...
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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: ...
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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 ...
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1answer
2k views

Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function

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) + ...
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1answer
202 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 ...
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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 ...
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2answers
460 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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2answers
178 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 ...
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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 ...
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2answers
61 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 ...
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0answers
141 views

Riemann vs. Ihara's $\zeta$ Function Variable Question

The Euler product for the Riemann zeta function $\zeta(z)$ implies that $$ \log\zeta_R(z)=\sum_{m>0}\frac{P(mz)}{m} \tag{R}, $$ whereas the Ihara zeta function for a graph $G$--all can be ...
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1answer
12 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$: ...
2
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1answer
71 views

A limit involving the Hurwitz zeta function

I want to show that $$ \lim_{s \to 1} \left( \zeta(s,a) - \frac{1}{s-1} \right) = - \psi(a)$$ where $\zeta(s,a)$ is the Hurwitz zeta function and $\psi(a)$ is the digamma function. The only ...
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1answer
28 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 ...
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1answer
283 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 ...
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2answers
61 views

Clausen zeta function

For $0 < \theta < 2\pi$, define $$\kappa(x,\theta) = \frac{1}{\zeta(x)}\sum_{n=1}^\infty \frac{e^{ in\theta}}{n^x}$$ for $\Re(x) > 1$. It is easy to see that $$\kappa(x,\theta) = ...
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1answer
67 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 ...
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1answer
49 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
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198 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 ...
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1answer
56 views

Generating function for the Hurwitz Zeta: $\sum_{k=1}^{\infty}\zeta_{H}(k+1,a)(-z)^{k}$

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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1answer
40 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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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 ...
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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
107 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 ...
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0answers
37 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
6 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.
4
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3answers
228 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 ...
5
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1answer
96 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 ...