On representations of error terms related to the derivatives for some Dirichlet series. (English) Zbl 1427.11093

Furuya, Jun; Minamide, T. Makoto; Tanigawa, Yoshio

Ill. J. Math. 61, No. 1-2, 187-209 (2017).

Summary: In previous papers, we examined several properties of an error term in a certain divisor problem related to the derivatives of the Riemann zeta-function. In this paper, we obtain representations of error terms related to the derivatives of some Dirichlet series, which can be regarded as generalized versions of a Dirichlet divisor problem and a Gauss circle problem. We also give the upper bounds of the error terms in terms of exponent pairs.

MSC:

11N37	Asymptotic results on arithmetic functions
11M06	\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

Dirichlet series; Gauss circle problem

References:

[1]	T. M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976. · Zbl 0335.10001
[2]	D. Banerjee and M. Minamide, The average behaviour of the error term in a new kind of the divisor problem, J. Math. Anal. Appl. 438 (2016), no. 2, 533-550. · Zbl 1382.11050 · doi:10.1016/j.jmaa.2016.02.013
[3]	X. Cao, J. Furuya, Y. Tanigawa and W. Zhai, A generalized divisor problem and the sum of Chowla and Walum, J. Math. Anal. Appl. 400 (2013), 15-21. · Zbl 1292.11106 · doi:10.1016/j.jmaa.2012.11.027
[4]	X. Cao, J. Furuya, Y. Tanigawa and W. Zhai, A generalized divisor problem and the sum of Chowla and Walum II, Funct. Approx. Comment. Math. 49 (2013), 159-188. · Zbl 1293.11097 · doi:10.7169/facm/2013.49.1.10
[5]	X. Cao, Y. Tanigawa and W. Zhai, On a conjecture of Chowla and Walum, Sci. China Math. 53 (2010), no. 10, 2755-2771. · Zbl 1266.11101 · doi:10.1007/s11425-010-4013-8
[6]	S. Chowla, Contributions to the analytic theory of numbers, Math. Z. 35 (1932), 279-299. · Zbl 0004.10202 · doi:10.1007/BF01186560
[7]	J. Furuya, M. Minamide and Y. Tanigawa, Representations and evaluations of the error term in a certain divisor problem, Math. Slovaca 66 (2016), 575-582. · Zbl 1389.11125 · doi:10.1515/ms-2015-0160
[8]	J. Furuya, M. Minamide and Y. Tanigawa, On a new circle problem, J. Aust. Math. Soc. 103 (2017), 231-249. · Zbl 1430.11132 · doi:10.1017/S1446788716000525
[9]	J. Furuya and Y. Tanigawa, On integrals and Dirichlet series obtained from the error term in the circle problem, Funct. Approx. Comment. Math. 51 (2014), no. 2, 303-333. · Zbl 1358.11106 · doi:10.7169/facm/2014.51.2.5
[10]	S. W. Graham and G. Kolesnik, Van der Corput’s method for exponential sums, London Mathematical Society Lecture Note Series, vol. 126, Cambridge University Press, Cambridge, 1991. · Zbl 0713.11001
[11]	D. R. Heath-Brown, The Piateckiǐ-Šapiro prime number theory, J. Number Theory 16 (1983), 242-266. · Zbl 0513.10042 · doi:10.1016/0022-314X(83)90044-6
[12]	M. N. Huxley, Exponential sums and lattice points III, Proc. Lond. Math. Soc. 87 (2003), 591-609. · Zbl 1065.11079 · doi:10.1112/S0024611503014485
[13]	A. Ivić, The Riemann zeta-function: Theory and applications, Dover Publications, Inc., Mineola, NY, 2003, Reprint of the 1985 original (Wiley). · Zbl 1034.11046
[14]	E. Krätzel, Lattice points, Kluwer Acad. Publishers, Dordrecht, 1988. · Zbl 0675.10031
[15]	M. Minamide, The truncated Voronoï formula for the derivative of the Riemann zeta function, Indian J. Math. 55 (2013), no. 3, 325-352. · Zbl 1295.11106
[16]	Y.-F. S. Pétermann, Divisor problems and exponent pairs, Arch. Math. 50 (1988), 243-250. · Zbl 0619.10034 · doi:10.1007/BF01187741