Multiplicative functions resembling the M?bius function,2
刘青阳 1吴杰2
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作者信息
1. 中国人民大学数学学院,北京 100872
2. CNRS LAMA 8050,Laboratoire d'Analyse et de Mathématiques Appliquées,Université Paris-Est Créteil,94010 Créteil Cedex,France
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摘要
设k≥2是一个正整数,f(n)是一个可乘算术函数,其Dirichlet级数F(s)具有如下形式:F(s)={L(s,x)H(s)ζ(ks)-1,若 k 为偶数,L(s,x)H(s)L(ks,x)-1,若 k 为奇数,其中,ζ 是Riemannζ-函数,x是一个模q≥3的非主Dirichlet特征,L(s,x)是相应的Dirichlet L-函数,且H(s)=∑∞n=1 h(n)n-s是在半平面Re s＞0上绝对收敛的Dirichlet级数,对σ＞0及某个常数A ≥ 0满足∑∞n=1|h(n)|n-σ(＜)σ-A.本文得到f(n)的部分和函数的上界估计及Ω-结果,这些结果改进并推广了 Aymone等(2022)的最新结果.
Abstract
Let k≥2 be an integer and let f(n)be a multiplicative function such that its Dirichlet series F(s)has the following representation:F(s)={L(s,x)H(s)ζ(ks)-1,2|k,L(s,x)H(s)L(ks,x)-1,2(ł)k,where ζ is the Riemann ζ-function,L(s,x)is the Dirichlet L-function associated with a non-principal Dirichlet character x modulo q ≥ 3,andH(s)=∑∞n=1 h(n)n-s is a Dirichlet series absolutely convergent for Res＞0 such that ∑∞n=1|h(n)|n-σ(＜)σ-A for all σ＞0 and some constant A ≥ 0.In this paper,we obtain an O-bound and an Ω-result for the summatory function of f(n),which improve some recent results of Aymone et al.(2022)concerning multiplicative functions resembling the Möbius function.
关键词
可乘函数/Möbius函数/Dirichlet级数
Key words
multiplicative function/Möbius function/Dirichlet series
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