Triple correlation sums of coefficients of cuspidal forms

Abstract

Triple correlation sums problem concerns the non-trivial power-saving bounds for the correlation of three objects. It is conjectured that these sums are non-trivial in any fixed but arbitrarily given ranges. In this paper, the uniform non-trivial bounds for the triple correlation sums ∑m≥1,n≥1λπ(1,m)λ⋆(n)λf(m+pn)U(m/X)V(n/H) in the level aspect are derived, where π is any GL3-Maaß cuspidal form, f∈Bk⁎(p), any Hecke newform of prime level p and weight k∈N+, and λ⋆(n), n≥1, are certain coefficients of arithmetic interest. As a result, we show that sums of this type follow the 1/3-significance level. We study the strength of the result by specifying ⋆ being the cusp forms on GL3 and GL2, respectively, and further obtain the more significant cancellations in these sums.