A relation to a remainder terms in an asymptotic formula for the associated Euler totient function
Hideto Iwata
TL;DR
This paper extends Montgomery's relation between error terms for the Euler totient to a degree-2 setting by analyzing the associated Euler totient function $\varphi(n,F)$ where $F$ is the $L$-function $L(s,f\\otimes\\chi)$ attached to a twisted Maass form. By leveraging the polynomial Euler product framework, Maass form theory, Hecke eigenvalues, and Perron's formula, the authors derive an analogue of Montgomery's error-term relation: $E_0(x,L)=x^{-1}E(x,L)+O\bigl(e^{-c_1\sqrt{\\log x}}\bigr)$. The proof hinges on expressing $\varphi(n,L)$ via auxiliary functions $\alpha(n)$ and $H(s)$, and showing the crucial identity $\frac{H(2,f\\otimes\\chi)}{L(2,f\\otimes\\chi)}=2C(L)$ to cancel main terms. The work provides a template for obtaining Montgomery-type relations for other degree-2 $L$-functions and highlights the interplay between automorphic $L$-functions and arithmetic error terms, with potential implications for precise asymptotics in generalized totient settings.
Abstract
H.L.Montgomery proved a relation for error terms in asymptotic formulas for the Euler totient function. J.Kaczorowski defined the associated Euler totient function which generalizes and obtained an asymptotic formula for it. In this paper, we prove a relation on error terms similar to H.L.Montgomery's result for a certain special case of the associated Euler totient function.