The Laplace transform of Dirichlet $L$-functions

A. Balčiūnas ; R. Macaitienė

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 86-96.

Résumé

Let $\chi$ be a Dirichlet character modulo $q$. The Dirichlet $L$-function $L(s,\chi)$ is defined in the half-plane $\sigma>1$ by the series $$ L(s,\chi)=\sum_{m=1}^{\infty}\frac{\chi(m)}{m^s}, $$ and has a meromorphic continuation to the whole complex plane. If $\chi$ is a non-principal character, then the function $L(s,\chi)$ is entire one. In the case of the principal character, the function $L(s,\chi)$ has unique simple pole at the point $s=1$. Dirichlet $L$- functions play an important role in the investigations of the distribution of prime numbers in arithmetical progresions, therefore, their analytic properties deserve a constant attention. In applications, often the moments of Dirichlet $L$-functions are used, whose asymptotic behaviour is very complicated. For investigation of moments, various methods are applied, one of them is based on the application of Mellin transforms. On the other hand, Mellin transforms use Laplace transforms. In the paper, the formulae for the Laplace transform of the function $\arrowvert L(s,\chi) \arrowvert^2$ in the critical strip are obtained. They extend the formulae obtained in [BaLa] on the critical line $\sigma=\frac{1}{2}$.

Détail

Keywords: Dirichlet $L$-function, Laplace transform, Mellin transform, Riemann zeta-function.

Bibliographie

[1] Atkinson A. A., “The mean value of the zeta-function on the critical line”, Quart. J. Math. Oxford, 10 (1939), 122–128 | DOI

[2] Apostol T. M., Introduction to Analytic Number Theory, Springer, Berlin, 1976 | MR | Zbl

[3] Balčiūnas A., Laurinčikas A., “The Laplace transform of Dirichlet L-functions”, Nonlinear Anal. Model. Control., 17 (2012), 127–138 | MR | Zbl

[4] Voronin A. A., Karatsuba A. A., Dzeta-funktsiya Rimana, Fizmatlit, M., 1994

[5] Ivič A., “The Voronoi identity via the Laplace transform”, Ramanujan J., 2 (1988), 39–45 | MR

[6] Ivič A., “The Laplace transform of the square in the circle an divisor problems”, Studia Sci. Math-Hung., 32 (1996), 181–205 | MR | Zbl

[7] Ivič A., “The Laplace and Mellin transforms of powers of the Riemann zeta-function”, Int. J. Math. Anal., 1–2 (2006), 113–140 | MR | Zbl

[8] Iwaniec H., Kowalski E., Analytic number theory, Amer. Math. Soc., Colloq. Publ., 53, 2004 | MR | Zbl

[9] Jutila M., “Mean values of Dirichlet series via Laplace transform, in Analytic number theory”, London Math. Soc. Lecture Note, 247, Cambridge Univ. Press, Cambridge, 1997, 169–207 | MR | Zbl

[10] Kačinskaitė R., Laurinčikas A., “The Laplace transform of the Riemann zeta-function in the critical strip”, Integral Transf. Spec. Funct., 20 (2009), 643–648 | DOI | MR

[11] Karatsuba A. A., Osnovy analiticheskoi teorii chisel, Nauka, M., 1983

[12] Lukkarinen M., The Mellin transform of the square of Riemann's zeta-function and Atkinson's formula, Ann. Acad. Sci. Fenn. Math. Diss., 140, Suomalainen Tiedeakatemia, Helsinki, 2005 | MR | Zbl

[13] Prachar K., Primzahlverteilung, Springer-Verlag, Götingen–Heidelberg–Berlin, 1957 | MR | Zbl

[14] Titchmarsh E. C., Theory of Functions, Oxford University Press, Oxford, 1939 | MR | Zbl

[15] Titchmarsh E. C., The Theory of the Riemann Zeta- Function, 2nd ed., revised by D. R. Heath-Brown, Clarendon Press, Oxford, 1986 | MR | Zbl