The generalized Bourgain-Sarnak-Ziegler criterion and its application to additively twisted sums on GL m
- PDF / 320,717 Bytes
- 24 Pages / 612 x 792 pts (letter) Page_size
- 69 Downloads / 184 Views
. ARTICLES .
https://doi.org/10.1007/s11425-020-1717-1
The generalized Bourgain-Sarnak-Ziegler criterion and its application to additively twisted sums on GLm Yujiao Jiang1,∗ & Guangshi L¨ u2 1School
of Mathematics and Statistics, Shandong University, Weihai 264209, China; 2School of Mathematics, Shandong University, Jinan 250100, China Email: [email protected], [email protected] Received March 14, 2020; accepted June 17, 2020
Abstract
The Bourgain-Sarnak-Ziegler (BSZ) criterion was initially established for an arbitrary multiplicative
function a(n) satisfying |a(n)| 6 1. Recently, Cafferata et al. (2020) showed that the condition |a(n)| 6 1 can be replaced by a(p) ≪ 1 to some extent. In this paper, we formulate and prove a further extension of the BSZ criterion, in which the restriction a(p) ≪ 1 is removed. As applications, we use it together with the analytic theory of automorphic L-functions to prove that there exist some cancellations in the sequence {λπ (n)e(nk α)}n>1 on GLm and the M¨ obius function is disjoint from this sequence. Keywords
the Bourgain-Sarnak-Ziegler theorem, exponential sums, automorphic L-function, Sarnak’s con-
jecture MSC(2010)
11N37, 11L07, 11F30, 11F66
Citation: Jiang Y J, L¨ u G S. The generalized Bourgain-Sarnak-Ziegler criterion and its application to additively twisted sums on GLm . Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-020-1717-1
1
Introduction
One of the basic problems in analytic number theory is to estimate sums over primes or correlations with the M¨obius function µ(n). One standard way to proceed is the bilinear method of Vinogradov, which is a technique to study sums over primes in terms of the type I sums and the type II sums. Later, Bourgain, Sarnak and Ziegler formulated a finite version of the bilinear method, which is the so-called Bourgain-Sarnak-Ziegler criterion, and is stated as follows. Theorem 1.1 (BSZ criterion [3]). Let a(n) be a multiplicative function with |a(n)| 6 1 and let ϕ : N → C with |ϕ| 6 1. Let τ > 0 be a small parameter and assume that for all distinct primes p, q 6 e1/τ , we have that for y large enough, ∑ 6 τ y. ϕ(pm)ϕ(qm) m6y
Then for x large enough,
∑ √ a(n)ϕ(n) 6 2 τ log(1/τ )x. n6x
* Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
math.scichina.com
link.springer.com
Jiang Y J et al.
2
Sci China Math
The BSZ criterion is a key ingredient in the arguments for many known cases of Sarnak’s conjecture (see [3,6,20,32], to list a few). It also plays an important role in investigating the sums related to multiplicative functions and trace functions (see, for example, [9, 19]). Recently, Cafferata et al. [5] established a generalization of the BSZ criterion, which includes multiplicative functions a(n) satisfying the Ramanujan conjecture a(p) ≪ 1. In this paper we first establish a further extension of the BSZ criterion that essentially covers multiplicative functions a(n) suitably bounded on average (see Hypothesis (a) below). Then we apply it to bound ce
Data Loading...
