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Mertens products in arithmetic progressions over function fields

Hwanyup Jung

TL;DR

The paper proves a function-field analogue of Mertens products in arithmetic progressions for primes in F_q[t], mirroring the integer case of Languasco–Zaccagnini. It uses a character decomposition and Weil’s Riemann hypothesis for Dirichlet L-functions over function fields to achieve square-root cancellation, yielding an unconditional GRH-strength asymptotic with no exceptional zeros. The main result shows a uniform Mertens-type formula P(n;Q,A_0) = C(Q,A_0) (n log q)^{-1/Φ(Q)} (1 + O(q^{-n/2})) for deg Q ≤ η n, with an explicit Euler-product constant C(Q,A_0). The approach highlights the function-field advantage, and the explicit constant encodes arithmetic data of the modulus and residue class, with potential extensions to Chebotarev-type splitting conditions in future work.

Abstract

We establish a function field analogue of Mertens' formula for Euler products restricted to primes in arithmetic progressions over the polynomial ring F_q[t]. Our results are in direct correspondence with those of Languasco and Zaccagnini for arithmetic progressions in the integers. Over function fields, Weil's Riemann hypothesis for Dirichlet L-functions holds unconditionally, and consequently the analogue of the GRH-strength asymptotic is obtained without any exceptional zero correction term.

Mertens products in arithmetic progressions over function fields

TL;DR

The paper proves a function-field analogue of Mertens products in arithmetic progressions for primes in F_q[t], mirroring the integer case of Languasco–Zaccagnini. It uses a character decomposition and Weil’s Riemann hypothesis for Dirichlet L-functions over function fields to achieve square-root cancellation, yielding an unconditional GRH-strength asymptotic with no exceptional zeros. The main result shows a uniform Mertens-type formula P(n;Q,A_0) = C(Q,A_0) (n log q)^{-1/Φ(Q)} (1 + O(q^{-n/2})) for deg Q ≤ η n, with an explicit Euler-product constant C(Q,A_0). The approach highlights the function-field advantage, and the explicit constant encodes arithmetic data of the modulus and residue class, with potential extensions to Chebotarev-type splitting conditions in future work.

Abstract

We establish a function field analogue of Mertens' formula for Euler products restricted to primes in arithmetic progressions over the polynomial ring F_q[t]. Our results are in direct correspondence with those of Languasco and Zaccagnini for arithmetic progressions in the integers. Over function fields, Weil's Riemann hypothesis for Dirichlet L-functions holds unconditionally, and consequently the analogue of the GRH-strength asymptotic is obtained without any exceptional zero correction term.
Paper Structure (8 sections, 5 theorems, 49 equations)

This paper contains 8 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Statement of the Main Theorem
  3. Analytic Preliminaries
  4. Character decomposition
  5. Dirichlet $L$--Functions and Square--Root Cancellation
  6. Tail estimates for twisted Euler products
  7. Proof of the Main Theorem
  8. The constant $C(Q,A_0)$

Key Result

Theorem 2.1

Fix $0<\eta<1$. Uniformly for all monic moduli $Q$ satisfying $\deg Q\le \eta n$ and all reduced residue classes $A_0 \bmod Q$, we have as $n\to\infty$, where $C(Q,A_0)>0$ is a constant depending only on the modulus $Q$ and the residue class $A_0$.

Theorems & Definitions (14)

  • Theorem 2.1: Mertens’ formula in arithmetic progressions
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1: Orthogonality
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Remark 4.1: On the shape of the error term
  • ...and 4 more