Mean value theorems for a class of Dirichlet series

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 23, Tome 357 (2008), pp. 201-223

Résumé

We obtain an asymptotic formula instead of the upper bounds of Chandrasekharan and Narasimhan (1964) and Lau (1999) for the mean square value of the error term associated with the Dedekind zeta-function of a cubic field $K_3$. We study also modular analogs of the classical divisor problems. Bibl. – 21 titles.

Bibliographie

[1] J. Kaczorowski, A. Perelli, “The Selberg class: a survey”, Number Theory in Progress, Vol. 2 (Zakopane–Kościelisko, 1997), de Gruyter, Berlin, 1999, 953–992 | MR | Zbl

[2] E. Landau, “Über die Anzahl der Gitterpunkte in gewissen Bereichen”, Gött. Nachr., 1912, 687–771

[3] A. de Roton, “On the mean square of the error term for an extended Selberg class”, Acta Arithm., 126 (2007), 27–55 | DOI | MR | Zbl

[4] K. Chandrasekharan, R. Narasimhan, “On the mean value of the error term of a class of arithmetical functions”, Acta Math., 112 (1964), 41–67 | DOI | MR | Zbl

[5] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd edn, revised by D. R. Heath-Brown, Clarendon Press, New York, 1986 | MR | Zbl

[6] A. Ivić, The Riemann zeta-function, Wiley, New York etc., 1985 | MR | Zbl

[7] Y.-K. Lau, “On the mean square formula of the error term for a class of arithmetical functions”, Monatsh. Math., 128 (1999), 111–129 | DOI | MR | Zbl

[8] M. N. Huxley, N. Watt, “The number of ideals in a quadratic field”, Proc. Indian Acad. Sci. Math. Sci., 104:1 (1994), 157–165 | DOI | MR | Zbl

[9] K. Chandrasekharan, R. Narasimhan, “The approximate functional equation for a class of zeta-functions”, Math. Ann., 152 (1963), 30–64 | DOI | MR | Zbl

[10] O. M. Fomenko, “Srednie znacheniya, svyazannye s dzeta-funktsiei Dedekinda”, Zap. nauchn. semin. POMI, 350, POMI, SPb., 2007, 187–198 | MR

[11] K.-C. Tong, “On divisor problems. I”, Acta Math. Sinica, 5 (1955), 313–324 ; “On divisor problems. III”, Acta Math. Sinica, 6 (1956), 515–541 | MR | Zbl | Zbl

[12] D. R. Heath-Brown, “The distribution and moments of the error term in the Dirichlet divisor problem”, Acta Arithm., 60 (1992), 389–415 | MR | Zbl

[13] A. Walficz, “Über die Koeffizientensummen einiger Modulformen”, Math. Ann., 108 (1933), 75–90 | DOI | MR

[14] J. L. Hafner, “On the representation of the summatory functions of a class of arithmetical functions”, Analytic Number Theory, Lecture Notes in Math., 899, Springer, Berlin, 1981, 148–165 | MR

[15] O. M. Fomenko, “Teoremy o srednikh znacheniyakh dlya avtomorfnykh $L$-funktsii”, Algebra i analiz, 19:5 (2007), 246–264 | MR

[16] A. Ivić, “Large values of certain number-theoretic error terms”, Acta Arithm., 56 (1990), 135–159 | MR | Zbl

[17] K. Matsumoto, “Liftings and mean value theorems for utomorphic $L$-functions”, Proc. London Math. Soc. (3), 90 (2005), 297–320 | DOI | MR | Zbl

[18] A. Good, “Approximative Funktionalgleichungen und Mittelwertsätze für Dirichletreihen, die Spitzenformen assoziiert sind”, Comment. Math. Helv., 50 (1975), 327–361 | DOI | MR | Zbl

[19] A. Good, “The square mean of Dirichlet series associated with cusp forms”, Mathematika, 29 (1982), 278–295 | DOI | MR | Zbl

[20] A. Ivić, “On zeta-functions associated with Fourier coefficients of cusp forms”, Proc. of the Amalfi Conf. on Analytic Number Theory (Maiori, 1989), Iniv. Salerno, Salerno, 1992, 231–246 | MR

[21] M. Jutila, Lectures on a method in the theory of exponential sums, Published for the Tata Institute of Fundamental Research, Bombay, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 80, Springer, Berlin, 1987 | MR