2022/01/10 by Zhi Qi, Qi, Zhi
Mathematics · Social Sciences · #11F12 #11F67 #Analytic Number Theory Research #FOS: Mathematics #Historical Geopolitical and Social Dynamics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2201.03198
openalex publication_date 2022/01/10 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
In this paper, we prove the following asymptotic formula for the spectral cubic moment of central L-values: ∑tf \leqslant T \frac 2 L ( \tfrac 1 2 , f )3 L(1, Sym2 f) + \frac 2 π ∫0T \frac | ζ (\tfrac 1 2 + it ) |6 | ζ (1 + 2 it ) |2 d t = T2 P3 (log T) + O (T1+ε) , where f ranges in an orthonormal basis of (even) Hecke--Maass cusp forms, and P3 is a certain polynomial of degree 3. It improves on the error term O (T8/7+ε) in a paper of Ivi'c and hence confirms his strong conjecture for the cubic moment. This is the first time that the (strong) moment conjecture is fully proven in a cubic case. Moreover, we establish the short-interval variant of the above asymptotic formula on intervals of length as short as Tε.