Celebratio Mathematica — Bateman

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P. T. Bateman, E. E. Kohlbecker, and J. P. Tull: “On a theorem of Erdős and Fuchs in additive number theory,” Proc. Am. Math. Soc. 14 : 2 (April 1963), pp. 278–284. MR 0144876 Zbl 0112.27405 article

Let \( a_1,a_2,\dots \) be an infinite sequence of integers such that \( 0\leq a_1\leq a_2\leq \cdots \) and \( a_k\to +\infty \). If \( n \) is a non-negative integer, let \( r(n) \) be the number of ordered pairs \( (i,j) \) of positive integers such that \( a_i + a_j \leq n \). Thus if \[ g(z) = \sum_{k=1}^{\infty}z^{a_k} = \sum_{n=0}^{\infty}c_nz^n \quad (|z| < 1), \] then \[ \sum_{n=0}^{\infty} r(n)z^n = g(z)^2/(1-z) \quad (|z| < 1). \] Erdős and Fuchs [1956] have shown that if \( c > 0 \), it is impossible to have \[ r(n) = cn + o(n^{1/4}(\log n)^{-1/2}) \] as \( n\to +\infty \). In this note we generalize their result to the case where the main term has the form \( nL(n) \), where \( L \) is a slowly oscillating function which is convex or concave on some interval of the form \( [a,+\infty) \).

Eugene Kohlbecker	Related
Jack Tull	Related