# Tagged Questions

The Riemann zeta function is defined as the analytic continuation of the function defined for σ > 1 by the sum of the preceding series.

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### 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 ...
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### The Zeta Function Before Riemann [duplicate]

Leonhard Euler studied the function that is now known as the Riemann zeta function. I have not found the notation $\zeta$ in any of the works of any mathematicians prior to Bernhard Riemann's paper On ...
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### Weil conjectures for higher dimensional cycles?

Let $X$ be a smooth projective variety over $\mathbb{F}_{q}$. For each pair of positive integers $n$ and $d$, let $\text{Chow}_{n,d}(X)$ denote the (coarse) moduli space of $n$-cycles of degree $d$ on ...
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### What does the higher coefficients of ihara zeta function reveal?

Assume we have a graph $G=(V,E)$. The ihara zeta function $Z(G,u)$ is of form $$\frac1{\displaystyle\sum_{i=0}^{2|E|}c_iu^i}$$ A graph which has $|E|$ edges cannot have a simple cycle of length ...
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### Weil Conjectures Analog for Multivariate Zeta Functions

We know that the Riemann zeta function can be generalized to multivariate zeta functions. Is there a multivariate analog of the Weil conjectures?
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### Zeros of the series $\phi(s, a) = \sum_{n \geq 0} e^{-an}(n + 1)^{-s}$

I have been recently interested in the series $\phi(s, a)$ of the title. There, a is defined to be any positive real number and s is a complex variable. The main reason for my curiosity is that the ...
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### Arithmetic zeta function and local zeta functions

For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product \zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s), ...
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### Good uses of Siegel zeros?

The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...
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### stable homotopy groups and zeta function

I have heard during a discussion that there is a well known relation between the stable homotopy groups of a sphere (more precisely the order of stable homotopy groups of localized sphere spectrum ...
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### Complex Geometry Consequesnces of Serre's Kahler-Zeta Function

Serre's famous paper Analogues K\"ahl\'eriens de Certaines Conjectures de Weil proves an analogue of the Weil conjectures for compact K\"ahler manifolds. It would go on to inspire the line of attack ...
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### Zeta-Determinant for shifted Laplacians on the circle

Consider on the circle $S^1$ the operator $$L := - \frac{\partial^2}{\partial \theta^2} + c$$ for some constant $c \in \mathbb{R}$. What is its $\zeta$-regularized determinant? This should be well-...
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### Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta

The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation. In the discrete calculus there is ...
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### Bound of Chebyshev function and zeros of zeta function

It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...
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### Analogues of the Riemann-Roch Theorem

In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that ...
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### Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants

Zeta functions abound in mathematics. Audrey Terras describes in Zeta Functions and Chaos three zeta functions--the zeta fct. of a projective non-singular algebraic variety; the Artin-Mazur zeta ...
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### A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where $$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$ is the Lerch zeta-...
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### How does one motivate the analytic continuation of the Riemann zeta function?

I saw the functional equation and its proof for the Riemann zeta function many times, but ususally the books start with, e.g. tricky change of variable of Gamma function or other seemly unmotivated ...
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### Tauberian theorem with better error term

This is a fairly vague question. Suppose we have a sequence of positive numbers $(c_n)_n$ and we want to find an asymptotic formula for $S(x) = \sum_{n \leq X} c_n$. In favorable circumstances, ...
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### Equivalence of Euler products of Dirichlet series and Meromorphic continuation

Suppose $(f_n(s))_n$ and $(g_n(s))_n$ are two sequences of Dirichlet series with positive coefficients such that $\exists \alpha\in\mathbb R$ such that for all $s\in\mathbb C$ with $\Re(s)>\alpha$ ...
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### Hasse-Weil L-Functions of CM Abelian Varieties

In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...
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### Explicit examples of Hasse--Weil zeta-function calculations for curves

The problem of calculating Hasse--Weil zeta-function for a given curve $C/\mathbb{F}_p$ over a finite field is far from being easy, especially for large genus (as discussed by Wouter Castryck at https:...
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### Riemann Siegel function and gamma function

I ask about an idea to prove this formula: $Γ(1/2-iβ)=((\sqrt{π})/(\sqrt{\coshπβ}))\exp(-i(2ϑ(β)+βln2π+\arctan(\tanh(1/2)πβ)))$ where $ϑ(β)$ is the Riemann Siegel function.
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### Functional equations for spectral zeta function

Functional equation in the theory of zeta functions is one of the important components of this theory. I am interested to know whether the similar property, having functional equation, for the ...