Omega theorems for the iterated additive convolution of a nonnegative arithmetic function

Let R_k(n) denote the number of ways of representing the integers not exceeding n as the sum of k members of a given sequence of nonnegative integers. Suppose that $\text{1}\text{2}\text{< β < k}$, $\text{δ =}\text{β}\text{2}\text{?}\text{β}\text{(4}\text{min}\text{(β,}\text{k}\text{2}\text{))}$ and

$\text{ξ=}\text{1/2β if β<k/2,}\text{β?1/2 if β=1/2,}\text{(k ? 2)(k + 1)/2k if k/2<β<k.}$

R. C. Vaughan has shown that the relation R_k(n) = G(n) + o(n^δ log^?ξn) as n → +∞ is impossible when G(n) is a linear combination of powers of n and the dominant term of G(n) is cn^β, c > 0. P. T. Bateman, for the case k = 2, has shown that similar results can be obtained when G(n) is a convex or concave function. In this paper, we combine the ideas of Vaughan and Bateman to extend the theorems stated above to functions whose fractional differences are of one sign for large n. Vaughan's theorem is included in ours, and in the case $\text{β <}\text{k}\text{2}$ we show that a better choice of parameter improves Vaughan's result by enabling us to drop the power of log n from the estimate of the error term.