A quantitative Erdös–Fuchs theorem and its generalization
Yong-Gao Chen, Min Tang · Acta Arithmetica · 2011
A quantitative Erdős-Fuchs theorem and its generalization by Yong-Gao Chen (Nanjing) and Min Tang (Wuhu) 1. Introduction.Let k ≥ 2 be a fixed integer and let A = {a 1 ≤ a 2 ≤ • • • } be an infinite sequence of nonnegative integers.We write F (z) = a∈A z a , A(n) = a∈A, a≤n 1 (counting repetitions).For n = 0, 1, 2, . . .let r k (A, n) denote the number of solutions ofIn 1956, Erdős and Fuchs [1] proved the following result:cannot hold for any constant c > 0.Jurkat (unpublished), and later Montgomery and Vaughan [5] improved the Erdős-Fuchs theorem by eliminating the log power on the right-hand side:cannot hold for any constant c > 0.Up to now, the Erdős-Fuchs theorem has been extended in various directions.For other related problems, see [2], [3], [4] and [6].Continuing this work, Tang [7] recently proved the following result.Theorem C. If A is an infinite sequence of nonnegative integers and k > 2, then r k (A, n) = cn + o(n 1/4 ) cannot hold for any constant c > 0.In this paper, we obtain a stronger version of the above results: 2010