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UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES

Part of: Multiplicative number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  17 July 2020

NICOLAS ROBLES  and
ARINDAM ROY Open the ORCID record for ARINDAM ROY [Opens in a new window]
NICOLAS ROBLES
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL61801, USA e-mail: nirobles@illinois.edu
ARINDAM ROY*
Affiliation:
Department of Mathematics and Statistics, UNC Charlotte, 9201 University City Blvd., Charlotte, NC28223, USA
*
e-mail: aroy15@uncc.edu
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Abstract

In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$, was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$, it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve.

Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$, where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$-free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$, $(a,q)=1$, uniformly on $q$. When $j\geq 2$, we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.

Keywords

Siegel–WalfiszSelberg’s theoremuniform asymptotic formulaDirichletL-functionWeil explicit formulageneralized Möbius functiongeneralized von Mangoldt function

MSC classification

Primary: 11M20: Real zeros of $L(s, chi)$; results on $L(1, chi)$ 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11N37: Asymptotic results on arithmetic functions
Secondary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 1 , August 2021 , pp. 127 - 144
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by I. Shparlinski

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