Exponential sums with multiplicative coefficients without the Ramanujan conjecture
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Mathematische Annalen
Exponential sums with multiplicative coefficients without the Ramanujan conjecture Yujiao Jiang1 · Guangshi Lü2 · Zhiwei Wang2 Received: 17 December 2019 / Revised: 18 October 2020 / Accepted: 20 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the exponential sum involving multiplicative function f under milder conditions on the range of f , which generalizes the work of Montgomery and Vaughan. As an application, we prove cancellation in the sum of additively twisted coefficients of automorphic L-function on GLm (m ≥ 4), uniformly in the additive character. Mathematics Subject Classification 11L07 · 11F30 · 11N36
1 Introduction Let M be the class of all complex valued multiplicative functions. For f ∈ M , the exponential sum involving multiplicative function f is defined by S(N , α) :=
f (n)e(nα),
(1.1)
n≤N
where e(t) = e2πit . The problem of estimating S(N , α) has attracted several mathematicians. Daboussi [8] first studied a class of 1-bounded multiplicative functions f ∈ F , where F ⊆ M denotes the set of those multiplicative functions f with | f (n)| ≤ 1. He proved that if
Communicated by Kannan Soundararajan.
B
Zhiwei Wang [email protected] Yujiao Jiang [email protected] Guangshi Lü [email protected]
1
School of Mathematics and Statistics, Shandong University, Weihai 264209, Shandong, China
2
School of Mathematics, Shandong University, Jinan 250100, Shandong, China
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Y. Jiang et al.
|α − a/q| ≤ 1/q 2 for some (a, q) = 1 and 3 ≤ q ≤ (N / log N )1/2 , then one has S(N , α)
N (log log N )1/2
(1.2)
uniformly for f ∈ F . An immediate corollary from Daboussi’s result (1.2) is that lim
N →∞
1 S(N , α) = 0 N
holds uniformly for all f ∈ F . A related problem is to characterize those functions f such that for every irrational α we have ⎞ ⎛ 1 1 S(N , α) = o ⎝ f (n)⎠ . N N n≤N We refer the reader to the excellent works of Dupain, Hall and Tenenbaum [10], Fouvry and Tenenbaum [12] for details. On the other hand, Montgomery and Vaughan [30] studied a more general class of multiplicative functions. More precisely, suppose that multiplicative function f satisfies the following two conditions: | f ( p)| ≤ A, for all primes p
(1.3)
and
| f (n)|2 ≤ A2 N , for all natural numbers N ,
(1.4)
n≤N
where A is an arbitrary constant with A ≥ 1. They proved that if |α − a/q| ≤ 1/q 2 for some (a, q) = 1 and 2 ≤ R ≤ q ≤ N /R, then S(N , α)
N N + 1/2 (log R)3/2 log N R
(1.5)
uniformly for f satisfying the conditions (1.3) and (1.4). As an application of (1.5), they obtained the following celebrated result concerning the upper bound of character sum: under the generalized Riemann hypothesis, one has
χ (n) q 1/2 log log q
n≤x
for any non-principal character χ modulo q. This estimate is essentially best possible.
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Later, Bachman [1] studied the upper bound of S(N , α) at various contexts. In particular, he improved the upper bound in (1.5) wi
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