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Hecke $L$-Functions and the Distribution of Totally Positive Integers

Published online by Cambridge University Press:  20 November 2018

Avner Ash  and
Solomon Friedberg
Avner Ash
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, U.S.A. email: ashav@bc.edu, friedber@bc.edu
Solomon Friedberg
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467-3806, U.S.A. email: ashav@bc.edu, friedber@bc.edu
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Abstract

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Let $K$ be a totally real number field of degree $n$. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

Keywords

11M4111F3011F5511H0611R47Eisenstein seriestoroidal integralFourier seriesHecke L-functiontotally positive integertrace
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 4 , 01 August 2007 , pp. 673 - 695
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Chinta, G. and Goldfeld, D., Grössencharakter L-functions of real quadratic fields twisted by modular symbols. Invent. Math. 144(2001), no. 3, 435–449.Google Scholar
[2] Duke, W. and Imamoḡlu, I., Lattice points in cones and Dirichlet series. Int. Math. Res. Not. (2004), no. 53, 2823–2836.Google Scholar
[3] Hecke, E., Über analytische Funktionen und die Verteilung von Zahlen mod.eins. Abh. Math. Seminar Hamburg 1(1921), pp. 54–76. Reprinted in: MathematischeWerke. Vandenhoeck & Ruprecht, Göttingen, 1983, pp. 313–335.Google Scholar
[4] Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen. Second edition. Chelsea Publishing, New York, 1953.Google Scholar
[5] Lang, S., Algebraic Number Theory. Addison-Wesley, Reading, MA, 1970.Google Scholar
[6] Piatetski-Shapiro, I. and Rallis, S., L-functions of automorphic forms on simple classical groups. In: Modular Forms, Ellis Horwood, Horwood, Chichester, 1984, pp. 251–261.Google Scholar
[7] Schnee, W., Über den Zusammenghang zwischen den Summabilitätseigenschaften Dirichletscher Reihen und irhem Funkionentheoretischen Charakter. Acta Math. 35(1912), 357–398.Google Scholar
[8] Siegel, C. L., Advanced Analytic Number Theory. Tata Institute of Fundamental Research Studies in Mathematics 9, Tata Institute of Fundamental Research, Bombay, 1980.Google Scholar