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Variance of primes in short residue classes for function fields

Stephan Baier, Arkaprava Bhandari

Abstract

Keating and Rudnick derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus $Q$ is a polynomial in $\mathbb{F}_q[T]$ such that $Q(0)\not=0$. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.

Variance of primes in short residue classes for function fields

Abstract

Keating and Rudnick derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus is a polynomial in such that . The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.
Paper Structure (8 sections, 5 theorems, 45 equations)

This paper contains 8 sections, 5 theorems, 45 equations.

Table of Contents

  1. Introduction to the problem
  2. Notations and fundamental relations
  3. Results
  4. Calculation of a mean value
  5. Treatment of the range $\deg Q >h$ (Proof of Theorem \ref{['maintheorem']}(i))
  6. Treatment of the range $\deg Q \le h+2$ (Proof of Theorem \ref{['maintheorem']}(ii))
  7. Equidistribution (Proof of Theorem \ref{['maintheorem']}(iii))
  8. Remarks on the case $Q(0)=0$

Key Result

Theorem 1

(i) Fix $n\ge 1$. Given a finite field $\mathbb{F}_q$, let $Q(T)\in \mathbb{F}_q[T]$ be a polynomial such that $\deg Q>n$. Then the implied constant being absolute. (ii) Fix $n\ge 2$. Given a sequence of finite fields $\mathbb{F}_q$ and square-free polynomials $Q(T)\in \mathbb{F}_q[T]$ of positive degree with $\deg Q\le n+1$, we have, as $q\rightarrow \infty$,

Theorems & Definitions (6)

  • Theorem 1: Theorem 2.2 in KR
  • Theorem 2: Theorem 2.1 in KR
  • Theorem 3: Variants of Corollaries 2.4. and 2.5. in BSR
  • Theorem 4
  • Theorem 5
  • Conjecture 1