A note on Mellin transform, Eisenstein Series and distribution dε_it on PSL(2,ℤ[i]) \backslash PSL(2,ℂ)

2019/12/27 by Otto Romero, Romero, Otto
Mathematics · #22E40 (Secondary) #58J51 (Primary) #Analytic Number Theory Research #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1912.12369

openalex publication_date 2019/12/27 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

Let f a smooth function with compact support defined on PSL(2,ℤ[i]) \backslash PSL(2,ℂ), we prove a formula for the Mellin transform of f, then we can define the micro-local lift dε_it to SL(2,ℂ). We calculate (f,dε_it) for f a cuspidal form and for f an incomplete Eisenstein series. We also establish asymptotic estimates when t tends to ∞. We conjecture that a new positive distribution dε_it^F, constructed with the Friedrichs' symmetrization technique, satisfies the same asymptotic estimates that dε_it. This would imply the quantum ergodicity for Eisenstein series on PSL(2,ℤ[i]) \backslash PSL(2,ℂ)

Related

• Mean Values and Quantum Variance for Degenerate Eisenstein Series of Higher Rank
• The Completed L-function attached to the Weight 2 Polar Harmonic Maass Form H_N,z^*(τ)
• Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions
• The modified Mellin transform of powers of the zeta-function
• The Riemann hypothesis via the Mellin transform, power series and the reflection relations