# Questions tagged [zeta-functions]

Questions on various generalizations of the Riemann zeta function (e.g. Dedekind zeta, Hasse–Weil zeta, L-functions, multiple zeta). Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.

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### Explore the relationship between $\sum\limits_{n = 1}^{2x} \frac{1}{{n^s}^x}$ and $\sum\limits_{n = 1}^{2x-1} \frac{(-1)^{n-1}}{{n^s}^x}$ [closed]

I am trying to find an algorithm with time complexity $O(1)$ for a boring problem code-named P-2000 problem. The answer to this boring question is a boring large number of $601$ digits. The DP ... 23 views

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### Zeta functions of hyperelliptic curves

I have been wondering recently about the geometric information encoded in the zeta function of a (smooth, projective) variety over a finite field - or in its étale cohomology (i.e. l-adic cohomology) ...
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### Analytic continuation of Dirichlet beta funcion

In terms of Hurwitz zeta function, Dirichlet beta function is given by $$\beta(s)=\frac1{4^s}\left(\zeta(s,\frac14)+\zeta(s,\frac34)\right).$$ Following the links, by means of analytic continuation of ...
1 vote
60 views

### Cancelling terms in an infinite series over $\mathbb{N}^2$

I try to understand Don Zagiers seemingly simple proof of the recurrence relation \begin{align*} \sum_{0<j<k,\ j\ \text{even}}\zeta(j)\zeta(k-j)=\frac{k+1}{2}\zeta(k)\ \ \ \ \ \ \ \ \ \ \ \ \ \...
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### Zeros of Riemann's $\xi(t)$

In Riemann's paper he defined $\xi(t)=\Pi(\frac{s}{2})(s-1)\pi^{-\frac{s}{2}}\zeta(s)$, where $s=\frac{1}{2}+ti$. On page 4 he said: The number of roots of $\xi(t)=0$, whose real parts lie between $0$...
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1 vote
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### A question from Titchmarsh's book " The Theory of the Riemann zeta function, Theorem 9.16, page 231

I am studying about upper bounds for $N(\sigma, T)$ (zero-density estimates), and while going through Theorem 9.16 in Titchmarsh's book (2nd edition) (page 231), I got a bit stuck in understanding the ...
76 views

### can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?

can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$? if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
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1 vote
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### Zero-free region of $\zeta^{(k)}(s)$

I am new to the Riemann zeta function. In the beginning of section 2 of Bruce c. Berndt, the number of zeros for $\zeta^{(k)}(s)$, J. London Math. Soc. (2), Vol 2, 1970, p. 577-580, the author made ...
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### Basic Riemann-zeta question (summability)

This is a quick question about when the sum of a product is the product of a sum. We have the definition of the zeta function as \begin{equation} \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}. \end{... 1 vote
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### Asymptotic behavior of $\psi^{(n)}(1/2+ix)$ as $n\to\infty$?

I'm trying to determine the behavior of an asymptotic series whose $n$th term (where $n$ is even, and all odd terms vanish) is proportional to $$\psi^{(n)}\left(\frac12+ix\right),$$ where $\psi^{(n)}$ ...
1 vote
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### Finding residues of a sum in paper by Hitchen

I am reading Nigel Hitchin's paper on Harmonic spinors [Adv. Math. 14 (1974) 1-55]. On p 34 he defines a function $$f(s)= \sum_{p,q\in{\mathbb Z}_+ }(p+q)(4pq \lambda^2 +(p-q)^2)^{-s}.$$ The ...