Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Introduction Read chapter Read first chapter

2013 | OriginalPaper | Chapter

2. Double Series of Bessel Functions and the Circle and Divisor Problems

Authors : George E. Andrews, Bruce C. Berndt

Published in: Ramanujan's Lost Notebook

Publisher: Springer New York

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This chapter is devoted to proofs of two remarkable identities involving infinite series of Bessel functions. One identity is associated with the classical circle problem, while the other is connected with the equally difficult divisor problem. Each of the identities is open to three different interpretations, with entirely different proofs needed for each interpretation. New methods of estimating trigonometric sums are introduced in our proofs.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Introduction
next chapter Koshliakov’s Formula and Guinand’s Formula
Literature
26.
B.C. Berndt, Identities involving the coefficients of a class of Dirichlet series. I., Trans. Amer. Math. Soc. 137 (1969), 345–359.
31.
B.C. Berndt, Identities involving the coefficients of a class of Dirichlet series, VII, Trans. Amer. Math. Soc. 201 (1975), 247–261.CrossRefMathSciNetMATH
44.
B.C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, Providence, RI, 2006.MATH
48.
B.C. Berndt, O.-Y. Chan, S.-G. Lim, and A. Zaharescu, Questionable claims found in Ramanujan’s lost notebook, in Tapas in Experimental Mathematics, T. Amdeberhan and V.H. Moll, eds., Contemp. Math., Vol. 457, American Mathematical Society, Providence, RI, 2008, pp. 69–98.
57.
B.C. Berndt, S. Kim, and A. Zaharescu, Weighted divisor sums and Bessel function series, II, Adv. Math. 229 (2012), 2055–2097.CrossRefMathSciNetMATH
58.
B.C. Berndt, S. Kim, and A. Zaharescu, Weighted divisor sums and Bessel function series, III, J. Reine Angew. Math., to appear.
59.
B.C. Berndt, S. Kim, and A. Zaharescu, Weighted divisor sums and Bessel function series, IV, Ramanujan J. 29 (2012), 79–102.CrossRefMathSciNetMATH
60.
B.C. Berndt, S. Kim, and A. Zaharescu, Circle and divisor problems, and double series of Bessel functions, Adv. Math. 236 (2013), 24–59.CrossRefMathSciNetMATH
71.
89.
K. Chandrasekharan and R. Narasimhan, Hecke’s functional equation and arithmetical identities, Ann. Math. (2) 74 (1961), 1–23.
90.
K. Chandrasekharan and R. Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. Math. (2) 76 (1962), 93–136.
101.
H. Davenport, Multiplicative Number Theory, 3rd ed., Springer-Verlag, New York, 2000.MATH
104.
C. Deninger, On the analogue of the formula of Chowla and Selberg for real quadratic fields, J. Reine Angew. Math. 351 (1984), 171–191.MathSciNetMATH
126.
I.S. Gradshteyn and I.M. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic Press, San Diego, 1994.
132.
A.P. Guinand, On Poisson’s summation formula, Ann. Math. (2) 42 (1941), 591–603.
139.
J.L. Hafner, Personal Communication, April 28, 2006.
144.
G.H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math. (Oxford) 46 (1915), 263–283.MATH
150.
G.H. Hardy, Collected Papers, Vol. II, Oxford University Press, Oxford, 1967.MATH
167.
C.G.J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Sumptibus, Fratrum Bornträger, Regiomonti, 1829.
269.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.MATH
288.
W. Sierpiński, O pewnem zagadnieniu z rachunku funkcyj asymptotycznych, Prace Mat. Fiz. 17 (1906), 77–118.
310.
M.G. Voronoï, Sur une fonction transcendante et ses applications à la sommation de quelques séries, Ann. École Norm. Sup. (3) 21 (1904), 207–267, 459–533.
314.
G.N. Watson, Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1966.MATH
Metadata
Title
Double Series of Bessel Functions and the Circle and Divisor Problems
Authors
George E. Andrews
Bruce C. Berndt
Copyright Year
2013
Publisher
Springer New York
Book
Ramanujan's Lost Notebook

Print ISBN: 978-1-4614-4080-2

Electronic ISBN: 978-1-4614-4081-9

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-4081-9_2

Premium Partner

    Image Credits