Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An Abel-Tauber theorem for partitions


Author: J. L. Geluk
Journal: Proc. Amer. Math. Soc. 82 (1981), 571-575
MSC: Primary 10J20; Secondary 26A12, 40E05
DOI: https://doi.org/10.1090/S0002-9939-1981-0614880-6
MathSciNet review: 614880
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $ \Lambda = \{ {\lambda _1},{\lambda _2}, \ldots \} $ is a given set of real numbers such that $ 0 < {\lambda _1} < {\lambda _2} < \ldots .{\text{Let }}n(u) = {\sum _{{\lambda _k} \leqslant u}}1$ and $ P(u)$ the number of solutions of $ {n_1}{\lambda _1} + {n_2}{\lambda _2} + \ldots \leqslant u$ in integers $ {n_i} \geqslant 0$. An Abel-Tauber theorem concerning $ n(u)$ and log $ P(u)$ is proved for the case where $ n(tx)/n(t) \to 1(t \to \infty )$ for $ x > 0$.


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

  • [1] A. A. Balkema, Monotone transformations and limit laws, Math. Centre Tracts 45, North-Holland, Amsterdam, 1973. MR 0334307 (48:12626)
  • [2] A. A. Balkema, J. L. Geluk and L. de Haan, An extension of Karamata's Tauberian theorem and its connection with complementary convex functions, Quart. J. Math. (2) 30 (1979), 385-416. MR 559046 (80m:40005)
  • [3] N. G. de Bruyn, Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform, Nieuw Arch. Wisk. (3) 7 (1959), 20-26. MR 0107120 (21:5847)
  • [4] -, On Mahler's partition problem, Nederl. Akad. Wetensch. Indag. Math. 10 (1948), 210-220.
  • [5] J. L. Geluk, On convolutions with the Möbius function, Proc. Amer. Math. Soc. 77 (1979), 201-210. MR 542085 (80j:40008)
  • [6] L. de Haan, On regular variation and its application to the weak convergence of sample extremes, Math. Centre Tracts 32, North-Holland, Amsterdam, 1970. MR 0286156 (44:3370)
  • [7] K. A. Jukes, Tauberian theorems of Landau-Ingham type, J. London Math. Soc. (2) 8 (1974), 570-576. MR 0345919 (49:10648)
  • [8] E. E. Kohlbecker, Weak asymptotic properties of partitions, Trans. Amer. Math. Soc. 88 (1958), 346-365. MR 0095808 (20:2309)
  • [9] E. G. H. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig-Berlin, 1909; reprint, Chelsea, New York, 1953.
  • [10] K. Mahler, On a special functional equation, J. London Math. Soc. (1) 15 (1940), 115-123. MR 0002921 (2:133e)
  • [11] S. Parameswaran, Partition functions whose logarithms are slowly oscillating, Trans. Amer. Math. Soc. 100 (1961), 217-240. MR 0140498 (25:3918)
  • [12] S. L. Segal, On convolutions with the Möbius function, Proc. Amer. Math. Soc. 34 (1972), 365-372. MR 0299572 (45:8620)
  • [13] -, A general Tauberian theorem of Landau-Ingham type, Math. Z. 111 (1969), 159-167. MR 0249379 (40:2624)
  • [14] -, Addendum to Jukes' paper on Tauberian theorems of Landau-Ingham type, J. London Math. Soc. (2) 9 (1974), 360-362. MR 0361521 (50:13966)
  • [15] E. Seneta, Regularly varying functions, Lecture Notes in Math., vol. 508, Springer-Verlag, 1976. MR 0453936 (56:12189)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10J20, 26A12, 40E05

Retrieve articles in all journals with MSC: 10J20, 26A12, 40E05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0614880-6
Keywords: Abel-Tauber theorems, regular variation, partition function
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society