Celebratio Mathematica — Bateman

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Works connected to Eugene Kohlbecker

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E. Landau: Elementary number theory. Chelsea (New York), 1958. English translation; with added exercises by P. T. Bateman and Eugene Kohlbecker. MR 0092794 Zbl 0079.06201 book

Eugene Kohlbecker	Related
Edmund Georg Hermann Landau	Related

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