Abstract
Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. As main results, this involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We establish these by the relation between bases of the Kubert space of functions. Then these expressions are equated with other expressions in terms of special functions introduced by some difference equations, giving rise to analogues of the Lerch-Chowla-Selberg formula. We also state Abelian results which not only yield asymptotic formulas for weighted summatory function from that for the original summatory function but assures the existence of the limit expression for Laurent coefficients.
Highlights
Whole mathematics especially number theory has been centered around the Laurent coefficients of the relevant zeta-and L-functions since Dirichlet, Kronecker, Lerch, Hecke, Siegel, Choela-Selberg
References [8,22] are another summit of the study on periodic Dirichlet series, appealing to the Deninger-Meyer method, based on the theory of Lerch zeta-function
Under the assumption of the functional equation, this depends on [25] and partly [26] and without it, depends on [23]. These two theorems pave a promenade to the study on Laurent series coefficients of a large class of zeta- and special functions starting from the Abelian theorem
Summary
Whole mathematics especially number theory has been centered around the Laurent coefficients of the relevant zeta-and L-functions since Dirichlet, Kronecker, Lerch, Hecke, Siegel, Choela-Selberg. The generalized Euler constants γk ( a, M ) in (53) for an arithmetic progression is naturally a highlighted subject and after [4,9,11,21], Shirasaka [12] is a culmination providing the genuine generating function for them, based on the theory of Hurwitz zeta-function In another direction, References [8,22] are another summit of the study on periodic Dirichlet series, appealing to the Deninger-Meyer method, based on the theory of Lerch zeta-function. Under the assumption of the functional equation, this depends on [25] and partly [26] and without it, depends on [23] These two theorems pave a promenade to the study on Laurent series coefficients of a large class of zeta- and special functions starting from the Abelian theorem
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