Mean Values of Arithmetic Functions in Short Intervals and in Arithmetic Progressions in the Large‐degree Limit

Abstract

A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as large as possible. We study this problem in the function field setting, and prove for a wide class of arithmetic functions (namely factorization functions), that such an asymptotic result holds, allowing the size of the short interval to be as small as a square-root of the size of the full interval, and analogously for arithmetic progressions. For instance, our results apply for the indicator function of polynomials with a divisor of given degree, and are much stronger than those known for the analogous function over the integers. As opposed to many previous works, our results apply in the large-degree limit, where the base field F q is fixed. Our proofs are based on relationships between certain character sums and symmetric functions, and in particular we use results from symmetric function theory due to Egecioglu and Remmel. We also use recent bounds of Bhowmick, Lê and Liu on character sums, which are in the spirit of the Drinfeld-Vlȃduţ bound.

Square-root cancellation for sums of factorization functions over squarefree progressions in $\mathbb F_q[t]$

“…For θ < 1 2 , the same result was proven by Gorodetsky [12], with error term going to 0 as n goes to ∞ with fixed q, using the Riemann hypothesis. Thus Theorem 1.8 is a beyond the Riemann hypothesis analogue of [12,Theorem 1.2].…”

“…For θ < 1 2 , the same result was proven by Gorodetsky [12], with error term going to 0 as n goes to ∞ with fixed q, using the Riemann hypothesis. Thus Theorem 1.8 is a beyond the Riemann hypothesis analogue of [12,Theorem 1.2]. However, the cost for our result simultaneously going beyond Riemann and (unlike the results above) allowing an arbitrary factorization function is that our error term doesn't go to zero as n goes to infinity, but only goes to a small fraction.…”

“…For k = 2, we can do better than θ = 2 3 due to Hooley, Linnik, and Selberg, as soon as q > ⌊e 2 2 2 • 2 4 ⌋ = 472. For k = 3, we can do better than the value θ = 12 23 due to Fouvry, Kowalski, and Michel [11], as long as q > ⌊ e 2 •2 2 •12 2 •11 2 3 24 11 ⌋ = 386. For any k, to obtain a level of distribution beyond 1/2, we need q > e 2 (k − 1) 2 k 2 .…”

“…For f a monic polynomial over F q , expressed as r i=1 π e i i for distinct prime factors π i and exponents e i , let the extended factorization type of f , denoted ω f , be the multiset of pairs (deg π i , e i ). This notion was introduced by Gorodetsky [12].…”

“…Finally, in Section 9, we deduce Theorem 1.8 from Theorem 1.7. We handle each extended factorization type separately (in this aspect, the argument is similar to [12]). We detect a given extended factorization type using a sieve which is an alternating sum of terms where each term is a factorization function as in Definition 1.6.…”