Exponential Sums with Multiplicative Coefficients Without the Ramanujan Conjecture
Exponential sums with multiplicative coefficients without the Ramanujan conjecture Yujiao Jiang, Guangshi Lü, Zhiwei Wang To cite this version: Yujiao Jiang, Guangshi Lü, Zhiwei Wang. Exponential sums with multiplicative coefficients without the Ramanujan conjecture. Mathematische Annalen, Springer Verlag, 2021, 379 (1-2), pp.589-632. 10.1007/s00208-020-02108-z. hal-03283544 HAL Id: hal-03283544 https://hal.archives-ouvertes.fr/hal-03283544 Submitted on 11 Jul 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EXPONENTIAL SUMS WITH MULTIPLICATIVE COEFFICIENTS WITHOUT THE RAMANUJAN CONJECTURE YUJIAO JIANG, GUANGSHI LÜ, AND ZHIWEI WANG 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. 1. Introduction Let M be the class of all complex valued multiplicative functions. For f 2 M , the exponential sum involving multiplicative function f is defined by X S(N; α) := f(n)e(nα); (1.1) n6N where e(t) = e2πit.
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