Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Identities involving the coefficients of a class of Dirichlet series. V


Author: Bruce C. Berndt
Journal: Trans. Amer. Math. Soc. 160 (1971), 139-156
MSC: Primary 30.24; Secondary 10.00
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We derive various forms of the Voronoï summation formula for a large class of arithmetical functions. These arithmetical functions are generated by Dirichlet series satisfying a functional equation with certain gamma factors. Using our theorems, we establish several arithmetical identities.


References [Enhancements On Off] (What's this?)

  • [1] Bruce C. Berndt, Generalised Dirichlet series and Hecke’s functional equation, Proc. Edinburgh Math. Soc. (2) 15 (1966/1967), 309–313. MR 0225732
  • [2] Bruce C. Berndt, Arithmetical identities and Hecke’s functional equation, Proc. Edinburgh Math. Soc. (2) 16 (1968/1969), 221–226. MR 0250981
  • [3] -, Identities involving the coefficients of a class of Dirichlet series. I, Trans. Amer. Math. Soc. 137 (1969), 345-359. MR 38 #4656.
  • [4] Bruce C. Berndt, Identities involving the coefficients of a class of Dirichlet series. I, II, Trans. Amer. Math. Soc. 137 (1969), 345-359; ibid. 137 (1969), 361–374. MR 0236360, 10.1090/S0002-9947-1969-99934-2
  • [5] Bruce C. Berndt, Identities involving the coefficients of a class of Dirichlet series. III, Trans. Amer. Math. Soc. 146 (1969), 323–348. MR 0252330, 10.1090/S0002-9947-1969-0252330-1
  • [6] K. Chandrasekharan and Raghavan Narasimhan, Hecke’s functional equation and arithmetical identities, Ann. of Math. (2) 74 (1961), 1–23. MR 0171761
  • [7] K. Chandrasekharan and Raghavan Narasimhan, Functional equations with multiple gamma factors and the average order of arithmetical functions, Ann. of Math. (2) 76 (1962), 93–136. MR 0140491
  • [8] K. Chandrasekharan and Raghavan Narasimhan, An approximate reciprocity formula for some exponential sums, Comment. Math. Helv. 43 (1968), 296–310. MR 0249372
  • [9] A. L. Dixon and W. L. Ferrar, Lattice-point summation formulae, Quart. J. Math. Oxford Ser. 2 (1931), 31-54.
  • [10] -, Some summations over the lattice points of a circle. I, Quart. J. Math. Oxford Ser. 5 (1934), 48-63.
  • [11] -, Some summations over the lattice points of a circle. II, Quart. J. Math. Oxford Ser. 5 (1934), 172-185.
  • [12] -, On the summation formulae of Voronoï and Poisson, Quart. J. Math. Oxford Ser. 8 (1937), 66-74.
  • [13] W. L. Ferrar, Summation formulae and their relation to Dirichlet’s series, Compositio Math. 1 (1935), 344–360. MR 1556898
  • [14] W. L. Ferrar, Summation formulae and their relation to Dirichlet’s series II, Compositio Math. 4 (1937), 394–405. MR 1556983
  • [15] I. S. Gradšteĭn and I. M. Ryžik, Tables of integrals, series and products, 4th ed., Fizmatgiz, Moscow, 1963; English transl., Academic Press, New York, 1965. MR 28 #5198; MR 33 #5952.
  • [16] A. P. Guinand, A class of self-reciprocal functions connected with summation formulae, Proc. London Math. Soc. (2) 43 (1937), 439-448.
  • [17] -, Summation formulae and self-reciprocal functions, Quart. J. Math. Oxford Ser. 9 (1938), 53-67.
  • [18] -, Summation formulae and self-reciprocal functions. II, Quart. J. Math. Oxford Ser. 10 (1939), 104-118.
  • [19] -, Finite summation formulae, Quart. J. Math. Oxford Ser. 10 (1939), 38-44.
  • [20] A. P. Guinand, Integral modular forms and summation formulae, Proc. Cambridge Philos. Soc. 43 (1947), 127–129. MR 0017774
  • [21] Hans Hamburger, Über einige Beziehungen, die mit der Funktionalgleichung der Riemannschen 𝜉-Funktion äquivalent sind, Math. Ann. 85 (1922), no. 1, 129–140 (German). MR 1512054, 10.1007/BF01449611
  • [22] N. S. Koshliakov, Application of the theory of sum-formulae to the investigation of a class of one-valued analytical functions in the theory of numbers, Messenger Math. 58 (1929), 1-23.
  • [23] -, On Voronoï's sum-formula, Messenger Math. 58 (1929), 30-32.
  • [24] E. Landau, Vorlesungen über Zahlentheorie, Zweiter Band, Chelsea, New York, 1947.
  • [25] C. Nasim, A summation formula involving 𝜎_{𝑘}(𝑛),𝑘>1, Canad. J. Math. 21 (1969), 951–964. MR 0246841
  • [26] C. Nasim, On the summation formula of Voronoi, Trans. Amer. Math. Soc. 163 (1972), 35–45. MR 0284410, 10.1090/S0002-9947-1972-0284410-9
  • [27] T. L. Pearson, Note on the Hardy-Landau summation formula, Canad. Math. Bull. 8 (1965), 717–720. MR 0194406
  • [28] W. Sierpiński, O pewnem zagadnieniu z rachunku funkcyj asymptotycznych, Prace Math.-Fiz. 17 (1906), 77-118.
  • [29] E. C. Titchmarsh, Han-shu lun, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR 0197687
  • [30] 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.
  • [31] -, Sur la développement, à l'aide des fonctions cylindriques, des sommes doubles $ \Sigma f(p{m^2} + 2qmn + r{n^2})$, où $ p{m^2} + 2qmn + r{n^2}$ est une forme positive à coefficients entiers (verhandlungen des Dritten Internat. Math.-Kongr, Heidelberg), Teubner, Leipzig, 1905, pp. 241-245.
  • [32] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
  • [33] J. R. Wilton, On Dirichlet's divisor problem, Proc. Roy. Soc. A 134 (1931), 192-202.
  • [34] -, Voronoïs summation formula, Quart. J. Math. Oxford Ser. 3 (1932), 26-32.
  • [35] -, An approximate functional equation with applications to a problem of Diophantine approximation, J. Reine Angew. Math. 169 (1933), 219-237.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30.24, 10.00

Retrieve articles in all journals with MSC: 30.24, 10.00


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-71-99991-0
Keywords: Arithmetical function, Voronoï summation formula, arithmetical identities, functional equation with gamma factors
Article copyright: © Copyright 1971 American Mathematical Society