Celebratio Mathematica

[1]P. T. Bateman: “Proof of a conjecture of Grosswald,” Duke Math. J. 25 : 1 (1958), pp. 67–72. MR 0091970 Zbl 0080.03602 article

[2]P. T. Bateman and E. Grosswald: “On a theorem of Erdős and Szekeres,” Ill. J. Math. 2 : 1 (1958), pp. 88–98. MR 0095804 Zbl 0079.07104 article

[3]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

[4]P. T. Bateman: “Lower bounds for \( \sum h(m)/m \) for arithmetical functions \( h \) similar to real residue-characters,” J. Math. Anal. Appl. 15 : 1 (July 1966), pp. 2–20. Dedicated to H. S. Vandiver on his eighty-third birthday. MR 0199164 Zbl 0144.27803 article

[5]P. T. Bateman: “The Erdős–Fuchs theorem on the square of a power series,” J. Number Theory 9 : 3 (August 1977), pp. 330–337. MR 0498470 Zbl 0353.10041 article

[6]P. T. Bateman, P. Erdős, C. Pomerance, and E. G. Straus: “The arithmetic mean of the divisors of an integer,” pp. 197–220 in Analytic number theory (Philadelphia, PA, 12–15 May 1980). Edited by M. I. Knopp. Lecture Notes in Mathematics 899. Springer (Berlin), 1981. Dedicated to Emil Grosswald on the occasion of his sixty-eighth birthday. MR 654528 Zbl 0478.10027 incollection

Marvin Isadore Knopp	Related
Paul Erdős	Related
Carl Bernard Pomerance	Related
Ernst Gabor Straus	Related
Emil Grosswald	Related

[7]P. T. Bateman and H. G. Diamond: “On the oscillation theorems of Pringsheim and Landau,” pp. 43–54 in Number theory. Edited by R. P. Bambah, V. C. Dumir, and R. J. Hans-Gill. Trends in Mathematics. Birkhäuser (Basel), 2000. MR 1764795 Zbl 0964.11037 incollection

Rajinder Jeet Hans-Gill	Related
Ram Prakash Bambah	Related
Harold George Diamond	Related
Vishwa Chander Dumir	Related

Paul T. Bateman

Arithmetic functions and summatory functions