On shifted convolution sums of GL(3)-Fourier coefficients with an average over shifts

2025/10/17 by Ritwik Pal, Pal, Ritwik, Sampurna Pal +1
Mathematics · #11F66 (primary) #11M41 (secondary) #Advanced Algebra and Geometry #Advanced Harmonic Analysis Research #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2510.15799

openalex publication_date 2025/10/17 · openalex created_date 2025/10/21 · openalex updated_date 2026/07/28

Abstract

Let F be a Hecke-Maass cusp form for SL₃(ℤ) and A(m,n) be its normalized Fourier coefficients. Let V be a smooth function, compactly supported on [1,2] and satisfying V(y)^j ≪_j y^-j for any j ∈ ℕ ∪ \0\. In this article we prove a power-saving upper bound for the `average' shifted convolution sum ∑_h∑_nA(1,n)A(1,n+h)V((n)/(N))V((h)/(H)), for the range N^1/2-ε ≥ H ≥ N^{1/6+ ε}, for any ε >0. This is an improvement over the previously known range N^1/2-ε ≥ H ≥ N^{1/4+ ε}.

Citations

• The second moment of the GL₃ standard L-function on the critical line
• Second moment of degree three L-functions
• Mellin-Barnes Integrals: A Primer on Particle Physics Applications
• The circle method and bounds for L-functions - III: t-aspect subconvexity for GL(3) L-functions
• Averages of shifted convolutions of d₃(n)
• The central value of the Rankin-Selberg L-functions
• A Sieve Method for Shifted Convolution Sums
• Mass equidistribution for Hecke eigenforms

Related

• Averages of shifted convolution sums for GL(3) × GL(2)
• Average shifted convolution sum for GL(d₁)× GL(d₂)
• A shifted convolution sum for GL(3)× GL(2)
• Shifted convolution sums of GL₃ cusp forms with θ-series
• Shifted Convolution Sum for GL(3) × GL(2) with Weighted Average