Tagged Questions

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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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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GCD function and relation between Hurwitz and Riemann zeta function

Does anyone know how to show the following: $\sum_{k=1}^n\gcd(n,k)\zeta(s,\frac{k}{n})=\left(\sum_{k=1}^n\gcd(n,k)\right)\zeta(s)$
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$\zeta(2 + it) = \zeta(2-it)$

Let $\zeta(s)$ denote the Riemann zeta-function. Show that $\zeta(2 + it) = \zeta(2-it)$ for all real t. Give some hints how to do this one.Thanks in advance.
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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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On the sum of prime powers

Has anybody investigated the asymptotic growth rate of functions in the form of $$f(z,n)=\sum\limits_{p\le n}p^z$$ For $Re(z)\ge -1$. Of course $f(0,n)=\pi (n)$ has an ocean of research surrounding it,...
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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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Calculating $\pi$ via the $\zeta$ function?

I was fooling around, trying to come up with a rapid way to compute $\pi$. Then I remembered that we always have: $$\zeta(2n)=c\pi^{2n},$$ where $n$ is a positive integer ...
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Question about $\int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) \forall s\in \Bbb N$

what I found from messing around was $$\int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s)$$ $$s\in \mathbb{N}$$ when the partial sum is changed to an equivalent polynomial using Faulhaber's formula. ...
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Relations betweens Multizeta Values

I am studying Multizeta values at the moment and I found that at weight 5, the basis is given by $\zeta(5)$ and $\zeta(3)\zeta(2)$ in the literature. Solving all shuffle and stuffle relations using ...
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Moebius / Zeta function connections

Following on from this question, I include a plot of the slightly less clear, but far simpler mathematically Mertens function against $x$ to the power of Zeta Zero 1, where the correlation between the ...
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Show that $f$ is harmonic

Let us consider the function: $$f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right]$$ My ...
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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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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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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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Properties of Dedekind zeta function

Suppose $K$ is a quadratic field and $a_K(n)$ denotes the number of ideals in the ring of integers of $K$ whose norm is equal to $n$. Then I need to show that $$\sum_{n\leq x} a_K(n)=O(x).$$ Clearly ...
how to prove that $\zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt$
how to prove that $$\zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt:R(s)>1 , R(q)>0$$ where $\zeta(s,q)$ is Hurwitz zeta function