1
$\begingroup$

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 $h.$

$\endgroup$

1 Answer 1

1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.