Geodesic
Mean value of real Dirichlet characters using a double Dirichlet series
Canadian mathematical bulletin, Tome 66 (2023) no. 4, pp. 1135-1151

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DOI

We study the double character sum $\sum \limits _{\substack {m\leq X,\\m\mathrm {\ odd}}}\sum \limits _{\substack {n\leq Y,\\n\mathrm {\ odd}}}\left (\frac {m}{n}\right )$ and its smoothly weighted counterpart. 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.
DOI : 10.4153/S000843952300022X
Mots-clés : Analytic number theory, Dirichlet characters, multiple Dirichlet series, character sums 
Čech, Martin. Mean value of real Dirichlet characters using a double Dirichlet series. Canadian mathematical bulletin, Tome 66 (2023) no. 4, pp. 1135-1151. doi: 10.4153/S000843952300022X
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     author = {\v{C}ech, Martin},
     title = {Mean value of real {Dirichlet} characters using a double {Dirichlet} series},
     journal = {Canadian mathematical bulletin},
     pages = {1135--1151},
     year = {2023},
     volume = {66},
     number = {4},
     doi = {10.4153/S000843952300022X},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S000843952300022X/}
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