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# Questions tagged [zeta-functions]

Zeta functions are typically analogues or generalizations of the Riemann zeta function. Examples include Dedekind zeta functions of number fields, and zeta functions of varieties over finite fields. They are typically initially defined as formal generating functions, but often admit analytic continuations.

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### Positivity of the coefficients of Taylor series associated to the Riemann hypothesis

The question below relates to the paper "Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier. I'm asking it here because I am sure the ...
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### Derivative of the Riemann zeta function at $z=-2$

I have a question regarding the derivatives of the Riemann zeta function. It is known that $\zeta'(-1)=\frac{1}{12}-\ln A$, where $A$ is the Glaisher-Kinkelin constant (which is an elegant ...
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### Derivative of zeta at positive even integers

Is there a general formula that sums up all values of $ζ′(2n)$, such that $n\in\mathbb{N}$?
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### Reference request for literature on the following function--power counting zeta function

I'll start by writing the character of interest, and describing some properties of it, before I get to the meat of the question. Any help is greatly appreciated, even an offhand suggestion/comment/...
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### Can I calculate congruent zeta function of given hyperelliptic curve by hand?

How can I calculate the numerator of congruent zeta function of given hyperelliptic curve ? For example, let $C:y^2=(x^2+1)(x^4-8x^3+2x^2+8x+1)$. numerator of congruent zeta function mod$23$ of this ...
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### Have Bushnell and Reiner got it wrong?

Let $M$ be a finite-dimensional simple $\mathbb Q$ algebra and $\Lambda$ an order in $M$. Its zeta-function is defined as $$Z(s)=\sum_{I}|\Lambda/I|^{-s},$$ where the sum runs over all left ideals ...
1 vote
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### Eisenstein series evaluated at $2i$

Consider the real analytic Eisenstein series defined by $$E(z,s) := \sum_{\gamma\in\Gamma_\infty\setminus\Gamma} Im(\gamma z)^s$$ where as usual, $\Gamma=SL(2,\mathbb{Z})$ and $\Gamma_\infty$ is the ...
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### Are the zeroes of the finite characteristic zeta functions dense in $\left\{s\in\mathbb{C}\mid\mathfrak{Re}(s)=\frac{1}{2}\right\}$?

If $p$ is a prime, $n\in\mathbb{N}$ is a natural number and $C$ is a nonsingular curve over $\mathbb{F}_{p^{n}}$, the $\zeta$ function associated to $C\mid_{\mathbb{F}_{p^{n}}}$ is defined as \begin{...
1 vote
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### Class of spectral zeta functions whose analytic extension takes a particular form

In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
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We know that the Ramanujan sums $$c_k:=\sum_{h<k, \gcd(h,k)=1}\cos \frac{2h \pi}{k}$$ have the property that $$\sum_{n \ge 1} \frac{c_n}{n^s}=\zeta(s)^{-1},\quad \text{where \displaystyle\zeta(s):=... 13 votes 0 answers 696 views ### Growth of residues of 1/\zeta(s): conjectures? Let \rho range over the non-trivial zeroes of the Riemann zeta function. Let$$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| = \max_{|\Im \rho|\leq T} \frac{1}...
Lets consider two views of zeta functions of curves. For the following, let $\mathbb{F}_p$ be the field with $p$ elements where $p$ is prime, and let $\overline{\mathbb{F}_p}$ be the algebraic closure ...