Functional and approximate functional equations of the Dirichlet function

Explicit Approximate Functional Equations for Various Classes of Dirichlet Series

“…Such explicit inequalities should be useful to compute special values of Dirichlet Lfunctions. Indeed Paris and Cang [8] recently derived from Lavrik's version of the approximate functional equation for the Riemann zeta-function [5] an approximate formula with a very good accuracy, and illustrated it with numerical examples. Approximate functional equations also appear in the study of the moments of Riemann's zeta function (see for instance [2]).…”

“…Several authors studied the error term R(χ, s, X, Y ). Lavrik [5] proved that R(χ, s, X, Y ) = O(q X −σ log(Y + 2) + q 1/2 (qt) 1…”

An asymptotic representation for $\zeta (\frac{1}{2} + it)$

“…We prove (1.4) by a slight modification of Riemann's original analysis; alternative methods of establishing this result are described in [8,11,12]. When Re(s) > 1, Riemann obtained the representation…”

“…In [10], a different form of expansion of Z(£), which also involves the complementary error function, was given using the normalized incomplete gamma function Q where A^ denotes an arbitrary positive integer, X(s) = 2V-1 sin£™r(l -s) = ^^r%~\ 8) , (1.3) and the coefficients A n (s) involve the incomplete gamma functions Q(l-5,±27rm(JV+^)).…”

Inversion formula of Dirichlet polynomials and the approximate functional equation of Dirichlet's L -functions

“…The approximate functional equation of Dirichlets functions was firstly considered by Lavrik in [2], and then improved by Rane and Zheng in [3] and [6] respectively. In [5] of Wang a smoothly weighted version of the approximate functional equation was firstly given which yielded an error term sharper than the previous ones by a factor t À1a2 .…”