I no - American Mathematical Society

proceedings of the american mathematical

society

Volume 82, Number 4, August 1981

AN ABEL-TAUBERTHEOREM FOR PARTITIONS J. L. GELUK Abstract. Suppose A = {X,, X2, .. . } is a given set of real numbers such that 0 < X, < X2 < ... . Let n(u) = 2^ 0.

Introduction. Suppose A = {A,, A2, . .. } is a given set of real numbers such that 0 < A! < A2 < . . . . Let 0 = v0 < j», < . . . be the elements of the additive semigroup generated by A. Consider the weighted partition the generating function

function p(vm) defined by

OO

II (1 - *-*T* = C e— dP(u):= P(s), r=\

J0

where P(u) = 2,. 0. Letting n(u) = S{t;^. oo)/or all x > 0. (ii) There exists a nondecreasing function a(t) on (0, oo) with lim(_,w a(t) = c such

that for all x > 0 x

I no /

(f-»oo).

If c < oo iAen

fXLLÍ!ldt -log P(x)^ log T(\ + c)

-'0

t

(x^oo).

If c = oo /Ae« "* «(/) log P(7) = C ^p-dt Jn •'o

I

+ n(x) + o(n(x)),

wAere x, y —*oo ane/ v — x/i(jc) — xa(y). If c = oo a/u/ /Ae function n satisfies the relation n(xn(x)) — n(x) then n(x) — a(x) and jx(n(s)/s) ds — log P(x) — n(x) log n(x) (x -* oo).

Received by the editors August 8, 1979.

AMS (MOS) subjectclassifications(1970). Primary 40E05, 26A12. Key words and phrases. Abel-Tauber

theorems, regular variation, partition function. © 1981 American

Mathematical

0002-99 39/81 /0000-0362/$02.25

571 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Society

572

J. L. GELUK

In order to prove the Main Theorem we use the Lambert transform A of « which

is defined by »(*)-/

Jo

-rs-]-n(u)du, e

—i

supposing the integral exists. This transform arises naturally since

log P(s) = - 2

^(r)log(l - e-*>°) = - f

r-l

J0

log(l - e"») x0, and (ii) for some real s, xg'(x)(log xf is nondecreasing for x > xx, imply that xg'(x) is slowly varying and so g(x) G n with auxiliary function xg'(x). Remark 2. If L(x) ~ M(x) (x -» oo) where M(x) is nondecreasing, then the

condition sup^^ L(y) = 0(L(x)) holds. This is trivial, since L(y)/L(x) < max(^4, (1 + e)M(x))/L(x) < c for allj> < x, where L(x) < (1 + e)M(x) (x > x0), A = supJt 0 and and asymptotic fxx(L(u)/u) du

and n(x)/x is integrable on (0, R) for R > 0, ñ(l/x) = }* (L(u)/u) du + o(L(x)) (x —>oo), to a nondecreasing function. Then n(x) — L(x) + o(L(x))

(x -► oo).

Proof. Since ~Zm 0 (t -►oo), by

Theorem 15 in [2] (in this case L ~ n). This stronger result is not very useful however, since log P is expressed in terms of A and not in terms of n.

References 1. A. A. Balkema, Monotone transformations and limit laws, Math. Centre Tracts 45, North-Holland,

Amsterdam, 1973. 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. 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. 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. .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. 7. K. A. Jukes, Tauberian theorems of Landau-Ingham

type, J. London Math. Soc. (2) 8 (1974),

570-576. 8. E. E. Kohlbecker,

Weak asymptotic properties of partitions, Trans. Amer. Math. Soc. 88 (1958),

346-365. 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. 11. S. Parameswaran,

Partition functions whose logarithms are slowly oscillating, Trans. Amer. Math.

Soc. 100 (1961),217-240. 12. S. L. Segal, On convolutions with the Möbius function, Proc. Amer. Math. Soc. 34 (1972), 365-372. 13._, A general Tauberian theorem of Landau-Ingham type, Math. Z. Ill (1969), 159-167. 14. _,

Addendum to Jukes' paper on Tauberian theorems of Landau-Ingham

type, J. London

Math. Soc. (2) 9 (1974),360-362. 15. E. Seneta, Regularly varying functions, Lecture Notes in Math., vol. 508, Springer-Verlag,

Department

of Mathematics,

Erasmus University

of Rotterdam,

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Rotterdam,

Holland

1976.