Arithmetic zeta function

In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmetic zeta function is one of the most fundamental objects of number theory.

Definition

The arithmetic zeta function ${\zeta _{X}(s)}$ can be defined by a formula analogous to the Euler product for the Riemann zeta function. Let $X$ be a scheme of finite type over $Z$. An irreducible subscheme $x$ is called a closed point if its residue field $k(x)$ which is the quotient ring of the local ring of $X$ at $x$ by its maximal ideal is finite. Denote by $|k(x)|$ the number of elements of $k(x)$. Then

${\zeta _{X}(s)}=\prod _{x}({1-|k(x)|^{-s}})^{-1}$

This product is taken over all closed points $x$ of the scheme $X$.

If $X$ is a finite field with $q$ elements, then ${\zeta _{X}(s)}={\frac {1}{1-q^{-s}}}$.

If $X$ is the spectrum of the ring of integers, then ${\zeta _{X}(s)}$ is the Euler-Riemann zeta function.

If $X$ is the spectrum of the ring of integers of an algebraic number field, then ${\zeta _{X}(s)}$ is the Dedekind zeta function.

Thus, the arithmetic zeta functions generalises the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields.

We also have ${\zeta _{P^{1}(X)}(s)}={\zeta _{X}(s-1)}$.

For more elementary properties see the references below.

Main conjectures

Let $X$ be a regular irreducible equidimensional scheme proper over integers (characteristic zero case) or over a finite field (positive characteristic). Then

1. The arithmetic zeta function ${\zeta _{X}(s)}$ has a meromorphic continuation to the complex plane and satisfies a functional equation with respect to $s\to n-s$ where $n$ is the absolute dimension of $X$.

2 (generalized Riemann Hypothesis). The zeros of ${\zeta _{X}(s)}$ inside the critical strip $Re(s)\in [0,n]$ lie on the vertical lines $Re(s)=1/2,3/2,\dots ,$ and the poles of ${\zeta _{X}(s)}$ inside the critical strip $Re(s)\in [0,n]$ lie on the vertical lines $Re(s)=0,1,2,\dots$

3 (special values). The order of the zero or pole and the residue of ${\zeta _{X}(s)}$ at integer points inside the critical strip can be described by formulas involving important arithmetic invariants of $X$.

What is proved

Conjecture 1 is proved when $n=1$ and some few special cases when $n>1$ in characteristic zero and for all $n$ in positive characteristic.

Conjecture 2 is proved (Emil Artin, Helmut Hasse, André Weil, Alexander Grothendieck, Pierre Deligne) in positive characteristic for all $n$. It is not proved for any schemes in characteristic zero. The Riemann hypothesis is a partial case of Conjecture 2.

Conjecture 3 is known when $n=1$ and is known in few very special cases when $n>1$. The Birch and Swinnerton-Dyer conjecture is a partial case of Conjecture 3.

References

• François Bruhat (1963). Lectures on some aspects of p-adic analysis. Tata Institute of Fundamental Research.
• Fesenko, Ivan (2010), "Analysis on arithmetic schemes. II", Journal of K-theory, 5: 437–557
• Jean-Pierre Serre (1965). Zeta and L-functions. Harper and Row. {{cite book}}: Unknown parameter |book= ignored (help)
• Serre, Jean-Pierre (1970), "Facteurs locaux des fonctions zeta des varietés algébriques (définitions et conjectures)", Seminaire Delange-Pisot-Poitou, 1969/70, 19