View source for Erdős-Fuchs theorem

{{TEX|done}}{{MSC|11B34}}

A statement about the number of ways that positive integers can be represented as a sum of two elements of a given set, stating that the [[Average order of an arithmetic function|average order]] of this number cannot be close to being a [[linear function]].   Given a subset $A$ of the natural numbers let $r(n)$ denote the number of ways that a natural number $n$ can be expressed as the sum of two elements of $A$ (taking order into account).  We consider the average 
$$
R(n) = (r(1)+r(2)+\cdots+r(n) ) / n  \ .
$$
The theorem states that 
$$
R(n) = C + O\left(n^{-3/4-\epsilon}\right) 
$$
cannot hold unless $C=0$

====References====
* P. Erdős, W.H.J. Fuchs, ''On a Problem of Additive Number Theory''Journal of the London Mathematical Society '''31''' (1956) 67–73 {{DOI|10.1112/jlms/s1-31.1.67}} {{ZBL|0070.04104}}
* D.J. Newman, "Analytic number theory", Graduate Texts in Mathematics '''177''' Springer (1998) ISBN 0-387-98308-2 {{ZBL|0887.11001}}