Exponential sums with multiplicative coefficients without the Ramanujan conjecture

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 GL m (m 4), uniformly in the additive character.

Additive divisor problem for multiplicative functions

“…for Re s > 1. By the Rankin-Selberg theory and the inequality |λ π (n)| 2 ≤ λ π×π (n) for all positive integers n (see [21,Lemma 3.1]), one has…”

Correlations of multiplicative functions with automorphic L-functions

“…In order to apply our theorem 1.1 to the sums (2.1) and (2.2), we need to verify these two hypotheses. By the Rankin-Selberg theory and the inequality |λ π (n)| 2 ≤ λ π×π (n) for all positive integers n (see [16,Lemma 3.1]), one has…”

Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic L‐functions

“…Thus, when the function f retains reasonable decay in arithmetic progressions with modulus less than $$(\log N)^{2+\varepsilon }$$, we can expect that for any frequency $$\alpha \in \mathbb {T}$$, $$$$\begin{align} \frac{1}{N}\sum _{n\leqslant N}{\left(f(n)-\mathbb {E}_{[N]}f \right)}e(n\alpha )\ll (\log N)^{-1}, \end{align}$$$$where we have written $$\mathbb {E}_{[N]}f=\mathbb {E}_{n\in [N]}f(n)=\frac{1}{N}\sum _{n\leqslant N}f(n)$$ as the average of f on the discrete interval $$[N]=\lbrace 1,2,\dots ,N\rbrace$$. Recently, Jiang, Lü, and Wang [19] weaken the above condition (1.1) that f takes bounded values in primes to the following two conditions $$$$\begin{equation} \sum _{p\leqslant N}|f(p)|^2\log p\ll N; \end{equation}$$$$and …”

“…Then the twisted 𝐿-function is defined by𝐿(𝑠, 𝜋 × 𝜒) = ∏When 𝑝 ∤ 𝑞, we have{ 𝛼 𝑗,𝜋×𝜒 (𝑝) ∶ 1 ⩽ 𝑗 ⩽ 𝑚 } = { 𝛼 𝑗,𝜋 (𝑝)𝜒(𝑝) ∶ 1 ⩽ 𝑗 ⩽ 𝑚 } . 𝑗,𝜋 (𝑝)𝜒(𝑝) 𝑝 𝑠) −𝐿(𝑠, 𝜋 × 𝜒) ∏We also need the following convexity bound for 𝐿(𝑠, 𝜋 × 𝜒), see[19, Lemma 3.2],…”