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GENERALISED DIVISOR SUMS OF BINARY FORMS OVER NUMBER FIELDS

Part of: Multiplicative number theory

Published online by Cambridge University Press:  16 November 2017

Christopher Frei  and
Efthymios Sofos
Christopher Frei
Affiliation:
University of Manchester, School of Mathematics, Oxford Road, Manchester, M13 9PL, UK (christopher.frei@manchester.ac.uk)
Efthymios Sofos
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn, 53111, Germany (sofos@mpim-bonn.mpg.de)
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Abstract

Estimating averages of Dirichlet convolutions $1\ast \unicode[STIX]{x1D712}$, for some real Dirichlet character $\unicode[STIX]{x1D712}$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin’s conjecture for Châtelet surfaces. We introduce a far-reaching generalisation of this problem, in particular replacing $\unicode[STIX]{x1D712}$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin’s conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.

Keywords

number theorydivisor sumsbinary formssums of arithmetic functions

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11N56: Rate of growth of arithmetic functions 11N64: Other results on the distribution of values or the characterization of arithmetic functions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 1 , January 2020 , pp. 137 - 173
Copyright
© Cambridge University Press 2017 

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