Abstract
One fragment (p. 335) published with Ramanujan’s Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second, we prove a trigonometric analogue.
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Berndt, B.C.: Identities involving the coefficients of a class of Dirichlet series. I. Trans. Am. Math. Soc. 137, 345–359 (1969)
Berndt, B.C.: Character analogues of the Poisson and Euler–MacLaurin summation formulas with applications. J. Number Theory 7, 413–445 (1975)
Berndt, B.C., Zaharescu, A.: Weighted divisor sums and Bessel function series. Math. Ann. 335, 249–283 (2006)
Berndt, B.C., Kim, S., Zaharescu, A.: Weighted divisor sums and Bessel function series, II. Adv. Math. 229, 2055–2097 (2012)
Berndt, B.C., Kim, S., Zaharescu, A.: Weighted divisor sums and Bessel function series, III. J. Reine Angew Math., to appear
Berndt, B.C., Kim, S., Zaharescu, A.: The circle problem and a double series of Bessel functions (submitted for publication)
Chandrasekharan, K., Narasimhan, R.: Hecke’s functional equation and arithmetical identities. Ann. Math. (2) 74, 1–23 (1961)
Chandrasekharan, K., Narasimhan, R.: Functional equations with multiple gamma factors and the average order of arithmetical functions. Ann. Math. (2) 76, 93–136 (1962)
Davenport, H.: Multiplicative Number Theory, 3rd edn. Springer, New York (2000)
Gradshteyn, I.S., Ryzhik, I.M. (eds.): Table of Integrals, Series, and Products, 5th edn. Academic Press, San Diego (1994)
Hardy, G.H.: On the expression of a number as the sum of two squares. Q. J. Math. 46, 263–283 (1915)
Hardy, G.H.: Collected Papers, vol. II. Oxford University Press, Oxford (1967)
Hua, L.K.: Introduction to Number Theory. Springer, Berlin (1982)
Ramanujan, S.: The Lost Notebook and other unpublished papers, Narosa, New Delhi (1988)
Sierpinski, W.: O pewnem zagadnieniu z rachunku funkcyj asymptotycznych. Pr. Mat.-Fiz. 17, 77–118 (1906)
Voronoï, M.G.: Sur une fonction transcendante et ses applications à la sommation de quelques séries. Ann. Sci. Éc. Norm. Super. 21, 207–267, 459–533 (1904)
Watson, G.N.: Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1966)
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The first author’s research was partially supported by NSA Grant H98230-11-1-0200.
The third author’s research was partially supported by NSF Grant DMS-0901621.
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Berndt, B.C., Kim, S. & Zaharescu, A. Weighted divisor sums and Bessel function series, IV. Ramanujan J 29, 79–102 (2012). https://doi.org/10.1007/s11139-011-9352-8
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DOI: https://doi.org/10.1007/s11139-011-9352-8
Keywords
- Circle problem
- Bessel functions
- Riesz sums
- Weighted divisor sums
- Dirichlet L-series
- Ramanujan’s Lost Notebook