Average shifted convolution sum for GL(d₁)× GL(d₂)

2025/11/28 by Esrafil Ali Molla, Molla, Esrafil Ali
Mathematics · #Analytic Number Theory Research #Advanced Algebra and Geometry #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2511.23096

Abstract

We study the average shifted convolution sum B(H,N):= (1)/(H) ∑_{h ∼ H} ∑_{n ∼ N} A_π1(n) A_π2(n+h), where A_πi(n) denotes the Fourier coefficients of a Hecke--Maass cusp form π_i for SL(d_i,ℤ) with d_i≥ 4, i=1,2. We establish a nontrivial power-saving bound of B(H,N) for the range of the shift H≥ N^{1-(4)/(d₁+d₂)+ε} for any ε>0. For the cases d₁ = d₂ + 1 and d₁ = d₂, our result extends a result that can be derived from a theorem of Friedlander and Iwaniec. In particular, when d₁ = d₂, we reach the critical threshold H≥ N^1-2/d+ε such that any further improvement in this range yields a subconvexity bound for the corresponding standard L-function in the t-aspect.

Citations

• The second moment of the GL₃ standard L-function on the critical line
• On the Rankin–Selberg problem
• Averages of shifted convolutions of d₃(n)
• A Sieve Method for Shifted Convolution Sums

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