×

On integrals and Dirichlet series obtained from the error term in the circle problem. (English) Zbl 1358.11106

Summary: In this paper, we shall investigate several properties of integrals defined by \(\int_1^{\infty}t^{-\theta}P(t)\log^jtdt\) with a complex variable \(\theta\) and a non-negative integer \(j\), where \(P(x)\) is the error term in the circle problem of Gauss. We shall also study the analytic continuation of several types of the Dirichlet series related with the circle problem, and study a proof of the functional equation of the Dedekind zeta-function associated with the Gaussian number field \({\mathbb{Q}}(\sqrt{-1})\).

MSC:

11N37 Asymptotic results on arithmetic functions
11M32 Multiple Dirichlet series and zeta functions and multizeta values
11R42 Zeta functions and \(L\)-functions of number fields
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] R. Ayoub and S. Chowla, On a theorem of Müller and Carlitz , J. Number Theory 2 (1970), 342-344. · Zbl 0198.37501 · doi:10.1016/0022-314X(70)90062-4
[2] L. Carlitz, A formula connected with lattice points in a circle , Abh. Math. Sem. Univ. Hamburg 21 (1957), 87-89. · Zbl 0077.05105 · doi:10.1007/BF02941926
[3] A. Erdélyi, Higher Transcendental Functions , Vol. II, McGraw-Hill, New York, 1953. · Zbl 0052.29502
[4] J. Furuya and Y. Tanigawa, Analytic properties of Dirichlet series obtained from the error term in the Dirichlet divisor problem , Pacific J. Math. 245 (2010), no. 2, 239-254. · Zbl 1201.11090 · doi:10.2140/pjm.2010.245.239
[5] J. Furuya and Y. Tanigawa, Explicit representations of the integral containing the error term in the divisor problem , Acta Math. Hungar. 129 (2010), no. 1-2, 24-46. · Zbl 1266.11102 · doi:10.1007/s10474-010-9219-2
[6] J. Furuya and Y. Tanigawa, Explicit representations of the integrals containing the error term in the divisor problem II , Glasg. Math. J. 54 (2012), no. 1, 133-147. · Zbl 1269.11100 · doi:10.1017/S0017089511000474
[7] J. Furuya, Y. Tanigawa and W. Zhai, Dirichlet series obtained from the error term in the Dirichlet divisor problem , Monatsh. Math. 160 (2010), no. 4, 347-357. · Zbl 1296.11100
[8] S.W. Graham and G. Kolesnik, Van der Corput’s method of exponential sums , London Mathematical Society Lecture Note Series, 126, Cambridge University Press, 1991. · Zbl 0713.11001
[9] G.H. Hardy and E. Landau, The lattice points of a circle , Proc. Royal Soc. A 105 (1924), 244-258.
[10] M.N. Huxley, Exponential sums and lattice points III , Proc. London Math. Soc. (3) 87 (2003), no. 3, 591-609. · Zbl 1065.11079 · doi:10.1112/S0024611503014485
[11] A. Ivić, The Riemann Zeta-Function, Theory and applications , Reprint of the 1985 original (John Wiley & Sons, New York), Dover Publications, Inc., Mineola, NY, 2003. · Zbl 1034.11046
[12] A. Ivić, A note on the Laplace transform of the square in the circle problem , Studia Sci. Math. Hungar. 37 (2001), no. 3-4, 391-399. · Zbl 0996.11058
[13] S. Kanemitsu and R. Sita Rama Chandra Rao, On a conjecture of S. Chowla and of S. Chowla and H. Walum, I , J. Number Theory 20 (1985), 255-261. · Zbl 0467.10031 · doi:10.1016/0022-314X(85)90020-4
[14] I. Kátai, The number of lattice points in a circle , Ann. Univ. Sci. Budapest Rolando Eötvös, Sect. Math. 8 (1965), 39-60. (in Russian) · Zbl 0151.04401
[15] E. Krätzel, Lattice Points , Mathematics and its Applications (East European Series), Kluwer Academic Publishers Group, Dordrecht, 1988.
[16] S. Lang, Algebraic Number Theory , Second Edition, Graduate Texts in Mathematics 110, Springer-Verlag, New York, 1994. · Zbl 0811.11001
[17] S. Lang, Undergraduate Analysis , Second Edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. · Zbl 0962.46001
[18] C. Müller, Eine Formel der analytischen Zahlentheorie , Abh. Math. Sem. Univ. Hamburg 19 (1954), no. 1-2, 62-65. · Zbl 0057.28403 · doi:10.1007/BF02941554
[19] W.G. Nowak, Lattice points in a circle: an improved mean-square asymptotics , Acta Arith. 113 (2004), 259-272. · Zbl 1092.11039 · doi:10.4064/aa113-3-4
[20] E. Preissmann, Sur la moyenne quadratique du terme de reste du problème du cercle , C. R. Acad. Sci. Paris Sér. I 306 (1988), 151-154. · Zbl 0654.10042
[21] W. Recknagel, Varianten des Gauß schen Kreisproblems , Abh. Math. Sem. Univ. Hamburg 59 (1989), 183-189. · Zbl 0718.11047 · doi:10.1007/BF02942328
[22] D. Redmond, A generalization of a theorem of Ayoub and Chowla , Proc. Amer. Math. Soc. 86 (1982), 574-580. · Zbl 0505.10022 · doi:10.2307/2043587
[23] D. Redmond, Corrections and additions to “A generalization of a theorem of Ayoub and Chowla” , Proc. Amer. Math. Soc. 90 (1984), 345-346. · Zbl 0533.10038 · doi:10.2307/2045369
[24] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , (2nd ed. revised by D. R. Heath-Brown), Oxford University Press, Oxford, 1985. · Zbl 0042.07901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.