A note on arithmetical functions with absolutely convergent Ramanujan expansions (Q2816456)

Let \(f\) be an arithmetic function with a Ramanujan expansion \(f(n) = \sum \hat f(r) c_r(n)\). Assuming that the coefficients are decreasing at a rate \( \hat f(r) = O(r^{-1} (\log r)^{-\alpha})\) with \(\alpha>2\), the author proves an asymptotic formula for \(\sum_{n \leq N} f(n)\) with error term \(O\bigl( N (\log N)^{2-\alpha} \bigr)\). If two functions \(f,g\) satisfy such an assumption with \(\alpha>4\), then an asymptotic formula holds for \(\sum_{n \leq N} f(n)g(n+h) \) with error term \(O\bigl( N (\log N)^{4-\alpha} \bigr)\).