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A Tauberian theorem for partitions. (English) Zbl 0063.02973

The author’s principal aim is to deduce the asymptotic formulas \[ p(n)\sim e^{\pi\sqrt{2n/3}}/4\sqrt 3 n,\quad q(n) \sim e^{\pi\sqrt{n/3}}/4\cdot 3^{\frac14}n^{\frac34},\quad n\to\infty, \] from reasonably simple properties of the generating functions for \(p(n)\) and \(q(n)\), the number of unrestricted partitions of \(n\) and the number of partitions of \(n\) into unequal (or odd) parts, respectively. A Tauberian theorem for the integral \(f(s)= \int_0^\infty e^{-us}\,dA(u)\) is first proved. This can be thought of as an interpolation between the theorems of Hardy-Littlewood type which conclude \(A(u)\sim u^a\), and those of Wiener-Ikehara type which conclude \(A(u)\sim e^{au}\) as \(u\to\infty\). This general theorem is specialized to a theorem which is directly applicable to \(p(n)\), \(q(n)\) and other types of special partitions. In the application to \(p(n)\), for example, the result is obtained from an elementary knowledge of the asymptotic behavior of the generating function \(\sum_{n=0}^\infty = p(n)z^n\), when \(z=x+iy\) approaches the principal singularity \(z=1\) in an arbitrarily large “Stoltz angle”, \(\vert y\vert \le \Delta(1-x)\), \(0<\Delta<\infty\).

MSC:

11P82 Analytic theory of partitions
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