Abstract
We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors’ new proof of Halasz’s theorem on mean values to this simpler setting. Several of the technical difficulties that arise over the integers disappear in the function field setting, which helps bring out more clearly the main ideas of the proofs over number fields. We also obtain Lipschitz estimates showing the slow variation of mean values of multiplicative functions over function fields, which display some features that are not present in the integer situation.
Highlights
1 Introduction We begin by introducing multiplicative functions over the polynomial ring Fq[x], highlighting the analogy with multiplicative functions over the integers
1.1 An introduction to multiplicative functions over function fields In the polynomial ring Fq[x], where q is a prime power, let M denote the set of monic polynomials and let Mn denote the set of monic polynomials of degree n, so that |Mn| = qn
Let P denote the set of irreducible monic polynomials, and Pn those of degree n, and we reserve the letter P to denote irreducible monic polynomials
Summary
We begin by introducing multiplicative functions over the polynomial ring Fq[x], highlighting the analogy with multiplicative functions over the integers. The discrete relation (1.8) is replaced by the continuous integral equation uσ (u) = 0uχ (t)σ (u − t)dt, where χ (t) = ψ(yt)−1 n≤yt f (n) is an average of the multiplicative function f evaluated at prime powers (here y is a suitably large parameter), and σ (u) approximates (in many situations) the mean-value of the function f evaluated over integers up to yu. Such integral equations were first considered by Wirsing, and are discussed further in [3]
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