Riemann hypothesis for \(\mathbb{F}_ p[T]\)
type: Item
instance of: scholarly article
title: Riemann hypothesis for \(\mathbb{F}_ p[T]\)
zbMATH Open document ID: 0862.11040
author: Javier Diaz-Vargas
published in: Journal of Number Theory
publication date: 1997-05-26T00:00:00Z
review text: Let \(\mathbb{F}_r\), \(r= p^m\), be the finite field with \(r\) elements. Let \(N\) be a positive integer and let \(k\) be a nonnegative integer. Let \((n_0, \dots, n_k)\) be a \((k+1)\)-tuple of nonnegative integers. We write \(N=n_0 \oplus n_1 \oplus \cdots \oplus n_k\) if and only if the following are true: 1. \(N= \sum^k_{j=0} n_j\); 2. There is no carry over of \(p\)-adic digits in this sum; and 3. For \(0\leq j \leq k-1\) we have \(n_j>0\) and \(n_j \equiv 0 \pmod {r-1}\). Now let \({\mathbf A}: = \mathbb{F}_r [T]\) be the polynomial ring over \(\mathbb{F}_r\). Let \[ S_k(N): = \sum_{\substack{ a \in {\mathbf A}, a \text{monic}\\ \deg (a) = k}} a^N. \] In ``Finite sums and interpolation formulas over \(GF [p^n,x]\)'' [Duke Math. J. 15, 1001-1012 (1948; Zbl 0032.00303)] \textit{L. Carlitz} stated that \(S_k(N) \neq 0\) if and only if there exists a \((k+1)\)-tuple \((n_0, \dots, n_k)\) with \(N=n_0 \oplus n_1 \oplus \cdots \oplus n_k\); indeed, it is a straightforward exercise with Lucas' formula to show that this condition is a necessary one. Carlitz did not however give a proof of the sufficiency and only gave an inadequate sketch of the sufficiency of a result implying that the degree of \(S_k (N) \) arises from a certain unique \((k+1)\)-tuple satisfying the above conditions. In any case, this result of Carlitz, which seemed so innocuous for almost 50 years, is now turing out to be of fundamental importance for the arithmetic of polynomials. It is certainly yet another case of the amazing influence of Carlitz's old work on current research. The importance of Carlitz's statement is that it allows one to prove a certain ``Riemann hypothesis'' which naturally occurs in function field arithmetic. To see how this might come about we proceed as follows: first of all, we introduce a new variable \(x\) and set \[ z(x,-N): = \sum^\infty_{k=0} S_k (N)x^{-k}. \] It is easy to see from the binomial theorem, or Carlitz's result above, that the above formal power series…
Mathematics Subject Classification ID: 11G09
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11T55
zbMATH DE Number: 935008
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: Drinfeld modules
zbMATH Keywords: polynomial sums
zbMATH Keywords: arithmetic of polynomials
zbMATH Keywords: function field
zbMATH Keywords: special values
zbMATH Keywords: Newton polygons
zbMATH Keywords: Riemann zeta function
reviewed by: David Goss
MaRDI profile type: Publication
full work available at URL: jnth.1996.0100
OpenAlex ID: W1969140850
DOI: 10.1006/JNTH.1996.0100
description: scientific article; zbMATH DE number 935008