Corresponding to the lattice point problem for a random sphere Kendall and Rankin [8], Nakajima [9] considered the summatory function of the coefficients of the product of two Hurwitz zeta-functions and obtained the Bessel series expression. In this note we treat the case of the product of $\varkappa$ Hurwitz zeta-functions for an arbitrary integer $\varkappa\ge 2$ and obtain the expression in terms of the Voronoï-Steen function. This amounts to a refinement of corrected Nakajima’s formula with streamlining of the ambiguous argument.
References
D. Banerjee and J. Mehta, Linearized product of two Riemann zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 8, 123–126. 1310.11082 10.3792/pjaa.90.123 MR3266746 euclid.pja/1412341996
D. Banerjee and J. Mehta, Linearized product of two Riemann zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 8, 123–126. 1310.11082 10.3792/pjaa.90.123 MR3266746 euclid.pja/1412341996
K. Chakraborty, S. Kanemitsu and H. Tsukada, Vistas of special functions II, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. 1206.33001K. Chakraborty, S. Kanemitsu and H. Tsukada, Vistas of special functions II, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. 1206.33001
K. Chandrasekharan and R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93–136. MR140491 0211.37901 10.2307/1970267K. Chandrasekharan and R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93–136. MR140491 0211.37901 10.2307/1970267
K. Chandrasekharan and R. Narasimhan, On lattice-points in a random sphere, Bull. Amer. Math. Soc. 73 (1967), 68–71. MR211967 0154.30102 10.1090/S0002-9904-1967-11644-6 euclid.bams/1183528616
K. Chandrasekharan and R. Narasimhan, On lattice-points in a random sphere, Bull. Amer. Math. Soc. 73 (1967), 68–71. MR211967 0154.30102 10.1090/S0002-9904-1967-11644-6 euclid.bams/1183528616
H. Davenport, Multiplicative number theory, Lectures given at the University of Michigan, Winter Term, 1966. Lectures in Advanced Mathematics, No. 1, Markham Publishing Co., Chicago, Ill. (1967); 2nd ed. Springer, New York etc. 1980.H. Davenport, Multiplicative number theory, Lectures given at the University of Michigan, Winter Term, 1966. Lectures in Advanced Mathematics, No. 1, Markham Publishing Co., Chicago, Ill. (1967); 2nd ed. Springer, New York etc. 1980.
G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc., New York, 1964. 45.0387.03G. H. Hardy and M. Riesz, The general theory of Dirichlet's series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 18, Stechert-Hafner, Inc., New York, 1964. 45.0387.03
S. Kanemitsu and H. Tsukada, Contributions to the theory of zeta-functions, Series on Number Theory and its Applications, 10, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. 1311.11003S. Kanemitsu and H. Tsukada, Contributions to the theory of zeta-functions, Series on Number Theory and its Applications, 10, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. 1311.11003
D. G. Kendall and R. A. Rankin, On the number of points of a given lattice in a random hypersphere, Quart. J. Math., Oxford Ser. (2) 4 (1953), 178–189. 0052.14503 10.1093/qmath/4.1.178D. G. Kendall and R. A. Rankin, On the number of points of a given lattice in a random hypersphere, Quart. J. Math., Oxford Ser. (2) 4 (1953), 178–189. 0052.14503 10.1093/qmath/4.1.178
M. Nakajima, Shifted divisor problem and random divisor problem, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 3, 49–52. MR1222821 0786.11060 10.3792/pjaa.69.49 euclid.pja/1195511506
M. Nakajima, Shifted divisor problem and random divisor problem, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 3, 49–52. MR1222821 0786.11060 10.3792/pjaa.69.49 euclid.pja/1195511506
M. Nakajima, A new expression for the product of the two Dirichlet series. I, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 2, 19–22. MR1960737 1114.11072 10.3792/pjaa.79.19 euclid.pja/1116443602
M. Nakajima, A new expression for the product of the two Dirichlet series. I, Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 2, 19–22. MR1960737 1114.11072 10.3792/pjaa.79.19 euclid.pja/1116443602
S. W. P. Steen, Divisor functions: their differential equations and recurrence formulae, Proc. London Math. Soc. S2-31 (1930), no. 1, 47–80. MR1577480 56.1001.02 10.1112/plms/s2-31.1.47S. W. P. Steen, Divisor functions: their differential equations and recurrence formulae, Proc. London Math. Soc. S2-31 (1930), no. 1, 47–80. MR1577480 56.1001.02 10.1112/plms/s2-31.1.47
G. Voronoï, Sur une fonction transcendante et ses applications à la sommation de quelques séries (suite), Ann. Sci. École Norm. Sup. (3) 21 (1904), 459–533=Sob. Soc. II, Izd. Akad. Nauk Ukr. SSR, Kiev 1952, 51–165. 35.0220.01G. Voronoï, Sur une fonction transcendante et ses applications à la sommation de quelques séries (suite), Ann. Sci. École Norm. Sup. (3) 21 (1904), 459–533=Sob. Soc. II, Izd. Akad. Nauk Ukr. SSR, Kiev 1952, 51–165. 35.0220.01