# Asymptotic upper bounds for some convolution sums

Roman Holowinsky proved (see arXiv:0809.1640v3, Theorem 2, page 3) some nice asymptotic upper bounds for sums

$$S(d,x) = \sum_{1 \leq n \leq x} \vert f(n)g(n+d) \vert$$

for given multiplicative functions $f,g$ and given fixed integer $d$ with $0 < \vert d \vert \leq x.$

Question: What is known about the analogue convolution sums (that, however, do not seems to be a generalization of the above sums)

$$S(a,b,h,x) = \sum_{1 \leq n,m \leq x,\; an + bm=h} \vert f(n)g(m) \vert$$

for given multiplicative functions $f,g$ and for fixed positive integers $$a >0,\;b >0,$$ real $x$, and fixed appropriate integer $h.$

-