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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 $h.$

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There's a lot of work on such problems. One significant paper is Peter Shiu's A Brun Titchmarsh theorem for multiplicative functions in Crelle (1980). See also Mohan Nair's paper in Acta Arithmetica 1992 which is available at

http://matwbn.icm.edu.pl/ksiazki/aa/aa62/aa6234.pdf

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