An Abel-Tauber theorem for partitions
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- by J. L. Geluk PDF
- Proc. Amer. Math. Soc. 82 (1981), 571-575 Request permission
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
- A. A. Balkema, Monotone transformations and limit laws, Mathematical Centre Tracts, No. 45, Mathematisch Centrum, Amsterdam, 1973. MR 0334307
- 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. Oxford Ser. (2) 30 (1979), no. 120, 385–416. MR 559046, DOI 10.1093/qmath/30.4.385
- N. G. de Bruijn, Pairs of slowly oscillating functions occurring in asymptotic problems concerning the Laplace transform, Nieuw Arch. Wisk. (3) 7 (1959), 20–26. MR 107120 —, On Mahler’s partition problem, Nederl. Akad. Wetensch. Indag. Math. 10 (1948), 210-220.
- J. L. Geluk, An Abel-Tauber theorem on convolutions with the Möbius function, Proc. Amer. Math. Soc. 77 (1979), no. 2, 201–209. MR 542085, DOI 10.1090/S0002-9939-1979-0542085-7
- L. de Haan, On regular variation and its application to the weak convergence of sample extremes, Mathematical Centre Tracts, vol. 32, Mathematisch Centrum, Amsterdam, 1970. MR 0286156
- K. A. Jukes, Tauberian theorems of Landau-Ingham type, J. London Math. Soc. (2) 8 (1974), 570–576. MR 345919, DOI 10.1112/jlms/s2-8.3.570
- Eugene E. Kohlbecker, Weak asymptotic properties of partitions, Trans. Amer. Math. Soc. 88 (1958), 346–365. MR 95808, DOI 10.1090/S0002-9947-1958-0095808-9 E. G. H. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig-Berlin, 1909; reprint, Chelsea, New York, 1953.
- Kurt Mahler, On a special functional equation, J. London Math. Soc. 15 (1940), 115–123. MR 2921, DOI 10.1112/jlms/s1-15.2.115
- S. Parameswaran, Partition functions whose logarithms are slowly oscillating, Trans. Amer. Math. Soc. 100 (1961), 217–240. MR 140498, DOI 10.1090/S0002-9947-1961-0140498-X
- S. L. Segal, On convolutions with the Möbius function, Proc. Amer. Math. Soc. 34 (1972), 365–372. MR 299572, DOI 10.1090/S0002-9939-1972-0299572-2
- Sanford L. Segal, A general Tauberian theorem of Landau-Ingham type, Math. Z. 111 (1969), 159–167. MR 249379, DOI 10.1007/BF01111197
- S. L. Segal, Addendum to: “On Tauberian theorems of Landau-Ingham type” (J. London Math. Soc. (2) 8 (1974), 570–576) by K. A. Jukes, J. London Math. Soc. (2) 9 (1974/75), 360–362. MR 361521, DOI 10.1112/jlms/s2-9.2.360
- Eugene Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin-New York, 1976. MR 0453936
Additional Information
- © Copyright 1981 American Mathematical Society
- 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