Omega theorems related to the general Euler totient function

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:412

Article

Kaczorowski, Jerzy^1,2  Wiertelak, Kazimierz¹ 

[1] Adam Mickiewicz Univ, Fac Math & Comp Sci, PL-61614 Poznan, Poland

[2] Polish Acad Sci, Inst Math, PL-00956 Warsaw, Poland

关键词: Omega estimates;    Generalized Euler totient function;    Riemann zeta function;    Dirichlet L-functions;    Newforms L-functions;   

DOI  :  10.1016/j.jmaa.2013.10.050

来源: Elsevier

【 摘 要 】

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free