Finite Ramanujan expansions and shifted convolution sums of arithmetical functions

Abstract

For two arithmetical functions $f$ and $g$, we study the convolution sum of the form ${\sum }_{n\le N}f(n)g(n+h)$ in the context of its asymptotic formula with explicit error terms. Here we introduce the concept of finite Ramanujan expansion of an arithmetical function and extend our earlier works in this setup.

References in corpus (4)

• On the error term in a Parseval type formula in the theory of Ramanujan expansions
• On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals, III
• On the error term in a Parseval type formula in the theory of Ramanujan expansions II
• Symmetry and short interval mean-squares

Cited by in corpus (9)

• A map of Ramanujan expansions
• Finite and infinite Euler products of Ramanujan expansions
• Finite Ramanujan expansions and shifted convolution sums of arithmetical functions, II
• A smooth shift approach for a Ramanujan expansion
• Sieve functions in arithmetic bands, II
• Multiplicative Ramanujan coefficients of null-function
• A smooth summation of Ramanujan expansions
• The Hardy-Littlewood conjectures on the twin primes and the binary Goldbach problem are true
• Convergence of Ramanujan expansions, I [Multiplicativity on Ramanujan clouds]