Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2-barrier

Abstract

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class a we extend such averages out to moduli ≤ x 20 39 −δ .

When Does the Bombieri–vinogradov Theorem Hold for a Given Multiplicative Function?

“…An integer n is y-smooth if all of its prime factors are ≤ y. In [6] we proved the Bombieri-Vinogradov Hypothesis for y-smooth supported f ∈ C satisfying the Siegel-Walfisz criterion, provided y ≤ x 1/2−o (1) . For arbitrary f ∈ C, we may therefore use this result to obtain the Bombieri-Vinogradov Hypothesis for the f -values restricted to y-smooth n, and need a different approach for those n that have a large prime factor (that is, a prime factor > y).…”

Bombieri–vinogradov Theorems for Modular Forms and Applications

“…However, it is expected that a larger level of distribution (at least , as predicted by GRH) is admissible. It seems that a delicate adaptation of Huxley's method , Vaughan's argument , or the approach of Granville and Shao would lead to a resolution to the “low” level of distribution appearing in Theorems and .…”

“…However, it is expected that a larger level of distribution (at least 1 2 , as predicted by GRH) is admissible. It seems that a delicate adaptation of Huxley's method [22], Vaughan's argument [46,47], or the approach of Granville and Shao [16,17] would lead to a resolution to the "low" level of distribution appearing in Theorems 9 and 10. Furthermore, as remarked by Radziwiłł, in most cases, the "L-function approach" heavily relies on the automorphy of the related (multiplicative) arithmetic functions.…”

“…Moreover, Bombieri et al [4][5][6] proved (3) with a lager Q with an added proviso. Besides, several variants have been studied by Huxley [22], Murty-Murty [34], Murty-Petersen [35], and Granville-Shao [16,17] among many others.…”

Zeros of partial sums of L-functions

“…Harper and Soundararajan [10]. For the class of 1-bounded functions, namely C(1), recently Drappeau, Granville and Shao [12,13,7] also investigated the existence of Bombieri-Vinogradov type remainder estimates. In this paper, we develop new Halász-type results for the logarithmic mean values of functions in this class.…”