Abstract
Recently, Lalín, Rodrigue, and Rogers have studied the secant zeta function and its convergence. They found many interesting values of the secant zeta function at some particular quadratic irrational numbers. They also gave modular transformation properties of the secant zeta function. In this paper, we generalized secant zeta function as a Lambert series and proved a result for the Lambert series, from which the main result of Lalín et al. follows as a corollary, using the theory of generalized Dedekind eta-function, developed by Lewittes, Berndt, and Arakawa.
Highlights
Introduction e Dedekind eta-function and its limiting values have been considered by several authors starting from Riemann’s posthumous fragment [1] and Wintner [2] and later by Reyna [3] and Wang [4]. ere are many generalizations of the Dedekind eta-function as a Lambert series including those of Lewittes [5], Berndt [6], and Arakawa [7, 8]
We will introduce a generalization of the secant zeta function as a Lambert series
We introduce two Lambert series corresponding to (22) and
Summary
Found its special values at some particular quadratic irrational arguments. ey proved the following results. (2) When z is an algebraic irrational number and s ≥ 2. To prove this theorem, they have used the celebrated ue–Siegel–Roth theorem. Ey found the values of the secant zeta function at some quadratic irrational numbers. We will introduce a generalization of the secant zeta function as a Lambert series. Using the theory of generalized Dedekind eta-function due to Lewittes [5], Berndt [6], and Arakawa [7], we shall give a generalization of eorem 3. We begin by briefly describing the theory of generalized Dedekind eta-function, developed by Lewittes [5], Berndt [6], and Arakawa [7], which is a main tool in our study
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