A Generalization of the Secant Zeta Function as a Lambert Series

Recently, Laĺın, Rodrigue, and Rogers have studied the secant zeta function and its convergence. 1ey found many interesting values of the secant zeta function at some particular quadratic irrational numbers. 1ey 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 Laĺın et al. follows as a corollary, using the theory of generalized Dedekind etafunction, developed by Lewittes, Berndt, and Arakawa.


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]. In particular cases, they reduce to the cotangent or the cosecant zeta function. Lerch [9] in 1904 introduced the cotangent zeta function for an algebraic irrational number z and an odd positive integer s as He stated the following functional equation for the cotangent zeta function, but without proof.
Berndt [10], in 1973, focused on the cotangent zeta function for general s ∈ C and proved Lerch's functional equation for cotangent zeta function. He found many interesting explicit formulae for ξ(z, s) when z is a quadratic irrational and s ≥ 3 is an odd integer. One such pleasing formula is In fact, Berndt's work implies that � jξ( � j , s)π − s ∈ Q, where j is any positive integer and s ≥ 3 is an odd integer.

Secant Zeta Function
Recently, Lalín et al. [11] considered the secant zeta function ψ(z, s) :� ∞ n�1 sec(nπz) n s (5) and found its special values at some particular quadratic irrational arguments. ey proved the following results.
e series (5) is absolutely convergent in the following cases: (1) When z � p/q is a rational number with q odd and s > 1.
(2) When z is an algebraic irrational number and s ≥ 2.
To prove this theorem, they have used the celebrated ue-Siegel-Roth theorem.
Theorem 3 (see [11], eorem 3). Let E m denote the Euler numbers and let B m denote the Bernoulli numbers. Suppose that l is an even positive integer. en, for appropriate values of α, ey found the values of the secant zeta function at some quadratic irrational numbers. For j ∈ Z, After observing these values, they conjectured the following.
Conjecture 1 (see [11], Conjecture 1). If j is any positive integer and s is an even positive integer, then By a clever use of residue theorem, Berndt and Straub [12] proved the above functional equation (6), and from it they derived Furthermore, they connected the secant Dirichlet series with Eichler integrals of Eisenstein series and checked unimodularity of period polynomials. On the contrary, Charollais and Greenberg [13] related the secant Dirichlet series ψ(α, s) to the generalized eta-function which was studied by Arakawa [7]. ey proved that for s ∈ 2N, for all real quadratic irrationals α. ey used Arakawa's result to give an explicit formula for ψ(α, s) for real quadratic irrational numbers α.
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.

Work of Lewittes and Berndt
Lewittes and Berndt treat the case of the upper half-plane H while Arakawa treats the case of upper half plane limiting to an algebraic irrational number. Hereafter, we use the following notations: Lewittes [5] defined the generalization of the Dedekind eta-function as a Lambert series. For a pair (r 1 , r 2 ) of real numbers, z ∈ H and arbitrary s ∈ C, he considered the series A z, s, r 1 , r 2 :� m>− r 1 ∞ k�1 k s− 1 e kr 2 + k m + r 1 z , (12) where the first summation is over all integers m with m > − r 1 . He also introduced its associate as H z, s, r 1 , r 2 :� A z, s, r 1 , r 2 + e s 2 A z, s, − r 1 , − r 2 .
Example 1. For special choices of parameters r 1 and r 2 , the A-and H-functions reduce to the cosecant and cotangent zeta functions: Some more definitions will be required.
For any positive number λ, let I(λ, ∞) denote the integration path consisting of the oriented line segment (+∞, λ), the positively oriented circle of radius λ with center at the origin, and the oriented line segment (λ, +∞). Let for any pair (ω 1 , ω 2 ) of positive numbers and for z, t ∈ C. Berndt [6] proved the following transformation formula.

Work of Arakawa
Arakawa studied certain Lambert series associated to a complex variable s and an irrational real algebraic number α.
ose Lambert series are defined as limiting (boundary) values of the generalized Dedekind eta-functions studied by Berndt [6]. Arakawa obtained transformation formulae under the action of SL(2, Z) on those α.
For an irrational real algebraic number α and a pair (p, q) of real numbers, Arakawa [7] introduced a generalized eta-function defined as and its associate by Example 2. Again, if we consider (p, q) � (1/2, 0) and (p, q) � (1, 0), then also we will get the cosecant and cotangent zeta function: Mathematical Problems in Engineering where s ∈ C with R(s) < 0.
Consider the generalized eta-function , s ∈ C (26) corresponding to (22), for z ∈ H and a pair (p, q) ∈ R 2 with p > 0. en, one can see that this series is absolutely convergent for arbitrary s ∈ C. It can be easily checked that there is a link between the infinite series A(z, s, r 1 , r 2 ) and η(z, s, r 1 , r 2 ).

Lemma 1. For any pair
Now, from the definition of H-function (13), we have Hence, using Lemma 1, we get Similarly, we have

Lemma 2. For any algebraic irrational number α and a pair
Again by the definition of H-function (23)(due to Arakawa), we have erefore, by Lemma 2, we get Arakawa obtained the following transformation formulae for H(α, s, (p, q)), by virtue of eorem 4 of Berndt and Proposition 1.

Generalization of the Secant Zeta Function
We introduce two Lambert series corresponding to (22) and (12).

Lemma 4. Let α be an algebraic irrational number and
Proof. One can prove this result applying the ue-Siegel-Roth theorem, in a similar manner to Arakawa's procedure for proving the absolute convergence of the series η(α, s, p, q). □ Lemma 5. If z ∈ H and a pair (p, q) ∈ R 2 with p > 0, then the series η * (z, s, p, q) is absolutely convergent for any s ∈ C.
Proof. Using Lemmas 5 and 6, we can show that A * (z, s, r 1 , r 2 ) is absolutely convergent for s ∈ C.
□ Mathematical Problems in Engineering 7

Main Results
Consider the difference for each V from (40). Now, the second term in the above expression is the secant zeta function in view of (50). is difference is quite natural in the sense that it expresses the surplus after the modular transformation is applied.
We interpret the main result of Lalín et al. eorem 3 in this setting as a special case of for R(s) < 0, and locate it in a natural way as we will see in Corollary 1. Our main theorem is the following.
Theorem 7. For a real algebraic irrational α and a complex variable s with R(s) < 0, we have Also, where Φ k and Ψ k and (k � 0, 1, 2) are defined later. ey indicate the block of L-integrals and the block of H-functions, corresponding to the matrix V k , respectively. Also, Ω 0 is defined in (90).
e genesis of the transformation formula of Lalín et al. ( [11], eorem 3) for the secant zeta function is given by the sum of D * (V 1 ) and D * (V 2 ), which we have seen in Corollary 1. We will see in the proof of Corollary 1 that the term 2A * (α, s, 1/2, 0) on the left side and the secant zeta function on the right hand side naturally cancel each other. As this occurs only in such a pairing, this elucidates the hidden structure of the paired transformation formula from a more general standpoint.
Deduction of the Main eorem of Lalín et al. Firstly, we deduce eorem 3 from Corollary 1. To do that, let l � 2k be an even positive integer and s � 1 − l.
en, (63) amounts to is proves eorem 3. e following conjecture seems to be plausible.
two matrices in PSL 2 (Z) which are inverses to each other. en, for a pair (p, q) ∈ R 2 , can be expressible in terms of special values of the zeta and Lfunctions as we have seen for the sum of two explicit expressions for (66)

A * in Terms of A-and H-Functions
Before proving our main theorem we need to express A * in terms of A and H. We know that given a sum S � n a n with its even and odd parts S e and S o , where the even part is over all even integer values and odd part over odd integer values, the sum 2S e − S is the alternating sum n (− 1) n a n . Using this observation, we have the following result. Proof. By the definition of A * (z, s, r 1 , r 2 ), we have □ Mathematical Problems in Engineering 9 ere is a duplication formula for A(z, s, r 1 , r 2 ) which is as follows: A(z, s, r 1 , r 2 ) + A(z, s, r 1 , r 2 + 1/2) � 2 s A(2z, s, r 1 , 2r 2 ).
Proof. From Definition 1 of A(z, s, r 1 , r 2 ), we have A z, s, r 1 , r 2 + A z, s, r 1 , which follows from Examples 1 and 3, respectively.