Erdős–Fuchs theorem

In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.

The theorem is named after Paul Erdős and Wolfgang Heinrich Johannes Fuchs.

Statement

Let A be a subset of the natural numbers and 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 (1956). "On a Problem of Additive Number Theory". Journal of the London Mathematical Society 31 (1): 67-73.

• Donald J. Newman (1998). Analytic number theory, 31-38. ISBN 0-387-98308-2.