## 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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