Vertaisarvioitu alkuperäisartikkeli tai data-artikkeli tieteellisessä aikakauslehdessä (A1)
Mean value of real Dirichlet characters using a double Dirichlet series
Julkaisun tekijät: Čech Martin
Kustantaja: CAMBRIDGE UNIV PRESS
Julkaisuvuosi: 2023
Journal: Canadian Mathematical Bulletin
Tietokannassa oleva lehden nimi: CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES
Lehden akronyymi: CAN MATH BULL
Sivujen määrä: 17
ISSN: 0008-4395
eISSN: 1496-4287
DOI: http://dx.doi.org/10.4153/S000843952300022X
Verkko-osoite: https://doi.org/10.4153/S000843952300022X
Tiivistelmä
We study the double character sum (m <= X, m odd n)Sigma (n <= Y n odd)Sigma (m/n ) and its smoothly weighted counter-part. An asymptotic formula with power saving error term was obtained by Conrey, Farmer, and Soundararajan by applying the Poisson summation formula. The result is interesting because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages-it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.
We study the double character sum (m <= X, m odd n)Sigma (n <= Y n odd)Sigma (m/n ) and its smoothly weighted counter-part. An asymptotic formula with power saving error term was obtained by Conrey, Farmer, and Soundararajan by applying the Poisson summation formula. The result is interesting because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages-it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.
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