# Exponential sums with multiplicative coefficients without the Ramanujan conjecture

@article{Jiang2020ExponentialSW, title={Exponential sums with multiplicative coefficients without the Ramanujan conjecture}, author={Yujiao Jiang and Guangshi L{\"u} and Zhiwei Wang}, journal={Mathematische Annalen}, year={2020}, volume={379}, pages={589-632} }

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 $$\text {GL}_m$$ GL m $$(m\ge 4)$$ ( m ≥ 4 ) , uniformly in the additive character.

## 8 Citations

A Bombieri–Vinogradov Theorem for Higher-Rank Groups

- MathematicsInternational Mathematics Research Notices
- 2021

We establish a result of Bombieri–Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\pi $ of…

ON OF FOURIER COEFFICIENTS OF HECKE-MAASS CUSP FORMS ON GL 3

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- 2022

. We consider sign changes of Fourier coeﬃcients of Hecke-Maass cusp forms for the group SL 3 ( Z ). When the underlying form is self-dual, we show that there are ≫ ε X 5 / 6 − ε sign changes among…

Additive divisor problem for multiplicative functions

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- 2022

. Let τ denote the divisor function, and f be any multiplicative function that satisﬁes some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted…

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

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- 2021

We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that above functions are not correlated with finite degree polynomial…

On Signs of Fourier Coefficients of Hecke-Maass Cusp Forms on $\mathrm{GL}_3$

- Mathematics
- 2022

. We consider sign changes of Fourier coeﬃcients of Hecke-Maass cusp forms for the group SL 3 ( Z ). When the underlying form is self-dual, we show that there are ≫ ε X 5 / 6 − ε sign changes among…

A zero density estimate and fractional imaginary parts of zeros for $\mathrm{GL}_2$ $L$-functions

- Mathematics
- 2021

We prove an analogue of Selberg’s zero density estimate for ζ(s) that holds for any GL2 L-function. We use this estimate to study the distribution of the vector of fractional parts of γα, where α ∈ R…

Applications of zero-free regions on averages and shifted convolution sums of Hecke eigenvalues

- Mathematics
- 2021

. By assuming the Vinogradov-Korobov type zero-free regions and the generalized Ramanujan-Petersson conjecture, we give some nontrivial upper bounds of almost all short sums of Fourier coeﬃcients of…

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