MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $h.$

share|cite|improve this question

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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