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AFFINE CONVOLUTIONS, RAMANUJAN–FOURIER EXPANSIONS AND SOPHIE GERMAIN PRIMES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 1 / August 2023
- Published online by Cambridge University Press:
- 25 October 2022, pp. 11-18
- Print publication:
- August 2023
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- Article
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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.
