Abstract
Abstract
For a fixed integer h, the standard orthogonality relations for Ramanujan sums
$c_r(n)$
give an asymptotic formula for the shifted convolution
$\sum _{n\le N} c_q(n)c_r(n+h)$
. We prove a generalised formula for affine convolutions
$\sum _{n\le N} c_q(n)c_r(kn+h)$
. This allows us to study affine convolutions
$\sum _{n\le N} f(n)g(kn+h)$
of arithmetical functions
$f,g$
admitting a suitable Ramanujan–Fourier expansion. As an application, we give a heuristic justification of the Hardy–Littlewood conjectural asymptotic formula for counting Sophie Germain primes.
Publisher
Cambridge University Press (CUP)