Approximate Functional Equation for the Product of Functions and Divisor Problem

2005/02/07 by Vivek V. Rane, Rane, V. V.
Mathematics · #11M35 #FOS: Mathematics #Functional Equations Stability Results #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.math/0502126

openalex publication_date 2005/02/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For functions defined via Dirichlet/generalized Dirichlet series in some half planes of the complex plane, we give a new simple elementary approach to obtain an Approximate Functional Equation(AFE for short) for the product of functions with explicit remainder term. We do this,using a highly generalized identity following Motohashi's approach,wherein he uses a generalisation of Dirichlet's device.We use our method to give an AFE for the product of two Dirichlet L-series corresponding to primitive Dirichlet characters with hitherto unknown estimate for its remainder term.We also consideer AFEs of functions on the real line and the AFEs not only of the r-th power of Dirichlet L-series(or,of Riemann Zeta function)but also of the r-th root of it,where r is a positive integer

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