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.
1. Introduction
The 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]. There 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.
Theorem 1 (see [9]). For any algebraic irrational number z and sufficiently large positive integer , we havewherewhere is the i-th Bernoulli number.
Berndt [10], in 1973, focused on the cotangent zeta function for general and proved Lerch’s functional equation for cotangent zeta function. He found many interesting explicit formulae for when z is a quadratic irrational and is an odd integer. One such pleasing formula is
In fact, Berndt’s work implies that , where j is any positive integer and is an odd integer.
2. Secant Zeta Function
Recently, Lalín et al. [11] considered the secant zeta functionand found its special values at some particular quadratic irrational arguments. They proved the following results.
Theorem 2 (see [11], Theorem 1). The series (5) is absolutely convergent in the following cases:(1)When is a rational number with q odd and .(2)When z is an algebraic irrational number and .
To prove this theorem, they have used the celebrated Thue–Siegel–Roth theorem.
Theorem 3 (see [11], Theorem 3). Let denote the Euler numbers and let denote the Bernoulli numbers. Suppose that l is an even positive integer. Then, for appropriate values of α,
They found the values of the secant zeta function at some quadratic irrational numbers. For ,
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 to the generalized eta-function which was studied by Arakawa [7]. They proved that for ,for all real quadratic irrationals α. They used Arakawa’s result to give an explicit formula for 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 Theorem 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.
3. Work of Lewittes and Berndt
Lewittes and Berndt treat the case of the upper half-plane 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 of real numbers, and arbitrary , he considered the serieswhere the first summation is over all integers m with . He also introduced its associate as
Let . Put , then . Using the product definition of Dedekind eta-function , it is easy to show that
Let us see a couple of examples.
Example 1. For special choices of parameters and , the A- and H-functions reduce to the cosecant and cotangent zeta functions:Also,Some more definitions will be required.
Definition 1 (Hurwitz zeta function). For a positive number a, the Hurwitz zeta function
Definition 2. Let denote the characteristic function of integers, i.e.,For any positive number λ, let denote the integration path consisting of the oriented line segment , the positively oriented circle of radius λ with center at the origin, and the oriented line segment .
Letfor any pair of positive numbers and for .
Berndt [6] proved the following transformation formula.
Theorem 4 (see [6], Theorem 2). Let with . For any pair of real numbers, set . For with let . Then, for arbitrary , we havewhere
Here, is understood to be real-valued on the upper segment of .
4. Work of Arakawa
Arakawa studied certain Lambert series associated to a complex variable s and an irrational real algebraic number α. Those 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 on those α.
For an irrational real algebraic number α and a pair of real numbers, Arakawa [7] introduced a generalized eta-function defined asand its associate by
Example 2. Again, if we consider and , then also we will get the cosecant and cotangent zeta function:where with .
Theorem 5 (see [7], Lemma 1 and Theorem 2). Suppose and . Then, the infinite series is absolutely convergent if . If, in addition, and then has analytic continuation to , and the singularity at is at worst a simple pole.
Arakawa proved the absolute convergence of for , by using the Thue–Siegel–Roth theorem.
Consider the generalized eta-functioncorresponding to (22), for and a pair with . Then, one can see that this series is absolutely convergent for arbitrary . It can be easily checked that there is a link between the infinite series and .
Lemma 1. For any pair and , we have
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
Therefore, by Lemma 2, we get
Proposition 1 (see [7], Proposition 1). Let α be an irrational real algebraic number, and with . Let with . Set and . If , then
Arakawa obtained the following transformation formulae for , by virtue of Theorem 4 of Berndt and Proposition 1.
Theorem 6 (see [7], Theorem 1). Let α be any real algebraic irrational, and let with such that . For any pair of real numbers, set , and . Then, for ,where
Berndt [6] (p. 499) found the special values of at nonnegative integral arguments :where denotes the nth Bernoulli polynomial and .
Lemma 3 (see [7], Lemma 4). Let α be an irrational number in a real quadratic field and let be a pair of rational numbers. Then, there exist a totally positive unit β of and an element of which satisfy the conditions:(i)(ii)(iii)We choose such and , i.e., which satisfy the conditions of Lemma 3. Then, using condition (ii), we haveSince and , we can see easily from Theorem 6 that
Example 3. Let , and V as in Lemma 3 and with and . Then,Values at some particular matrices. Let
Example 4. Theorem 6 gives the following:In particular, when , we haveIf we choose , we getNote that for nonnegative integers m, we have the following explicit formulae for , where :
5. Generalization of the Secant Zeta Function
We introduce two Lambert series corresponding to (22) and (12). These include the generalizations of secant and tangent zeta functions as shown in Example 5. Let α be any algebraic irrational number and a pair of real numbers. Then, we define the series byand another infinite series byfor a pair , , and .
Example 5. If we take , and , then (48) becomesrespectively.
By virtue of the results of Lewittes, Berndt, and Arakawa, we have the following results.
Lemma 4. Let α be an algebraic irrational number and be a pair of real numbers. The series is absolutely convergent, if with .
Proof. One can prove this result applying the Thue–Siegel–Roth theorem, in a similar manner to Arakawa’s procedure for proving the absolute convergence of the series .
Lemma 5. If and a pair with , then the series is absolutely convergent for any .
Proof. Since , assume with . We havefor . And we can choose a large enough positive integer K such that for Thus,
Lemma 6. Let and α be an irrational algebraic number. Then, for any pair of real numbers , we have
Proof. If , then implies for . By the definition of , we knowAgain, if , implies for . So, we will haveSimilarly, we can see that
Lemma 7. If , is absolutely convergent for any .
Proof. Using Lemmas 5 and 6, we can show that is absolutely convergent for .
6. Main Results
Consider the differencefor each V from (40). Now, the second term in the above expression is the secant zeta function in view of (50). This 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. Theorem 3 in this setting as a special case offor 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 , we haveAlso,where and and () are defined later. They indicate the block of L-integrals and the block of H-functions, corresponding to the matrix , respectively. Also, is defined in (90).
We recover the main result of Lalín et al. ([11], Theorem 3), i.e., Theorem 3 by adding the equations (61) and (62). We note it as a corollary.
Corollary 1.
The genesis of the transformation formula of Lalín et al. ([11], Theorem 3) for the secant zeta function is given by the sum of and , which we have seen in Corollary 1. We will see in the proof of Corollary 1 that the term 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 Theorem of Lalín et al. Firstly, we deduce Theorem 3 from Corollary 1. To do that, let be an even positive integer and . Then, (63) amounts to
This proves Theorem 3.
The following conjecture seems to be plausible.
Conjecture 2. Let and be two matrices in which are inverses to each other. Then, for a pair ,can be expressible in terms of special values of the zeta and L-functions as we have seen for the sum of two explicit expressions for
7. A∗ in Terms of A- and H-Functions
Before proving our main theorem we need to express in terms of A and H. We know that given a sum with its even and odd parts and , where the even part is over all even integer values and odd part over odd integer values, the sum is the alternating sum . Using this observation, we have the following result.
Lemma 8.
Proof. By the definition of , we have
There is a duplication formula for which is as follows:
Lemma 9.
Proof. From Definition 1 of , we have
Using the duplication formula, i.e., Lemma 9 in Lemma 8, we get
Lemma 10.
On the other hand,
Therefore,
Substituting (71) in Lemma 10, we deduce the following proposition.
Proposition 2. For a real algebraic irrational α, a pair of real numbers with , and a complex variable s with , we havewhereas in equation (71).
Example 6. If we consider and , then we getFor the last term, with s an even integer, we use eitherorwhich follows from Examples 1 and 3, respectively.
8. General Procedure
The general procedure is to transform
We recall the following notations
Now using Proposition 2, we can write
For , we have
We now transform (81) by using (78):wherein the case of (79), whilein the case of (82). Hence,where the last term is either (79) or (85).
9. Proof of Theorem 7 (60)
The three identities in Theorem 7 are proved on similar lines. We begin by using (83) and (85).
Then, applying (41), we deduce that
Let
Now, we can express the difference as
Using the integral representation (35) of , we calculate . Therefore,
Combining the first two integrals, we havewhere
Now, making the change of variable , we get
Hence, after eliminating the common factor, we arrive at
Therefore,
Our next target is to calculate . Using (23), we have
Now, using the definition of the η-function, we get
To calculate , we use , and we get
Finally, combining the expressions for and we deduce Theorem 7 (60).
10. Proof of Theorem 7(61)
By using Proposition 2 and from (42), we have
Let
We now express (102) as
Now, utilizing the integral representation (35) of , we have
Again, we write the left hand side of the above equation aswhere
Now, by change of variable followed by the elimination of the common factor , we get
Thus, substituting in (107), we see that
Using the definition of H-function, from (104), we have
The nth summand isfrom which we may eliminate the common factor . Therefore,
Now, we substitute (110) and (113) in (102) and finally get
This completes the proof of Theorem 7 (61).
11. Proof of Theorem 7 (62)
We follow the same route: first, we use Proposition 2 and then using (43), we obtainwhere
To simplify , we make use of the integral representation (35) of . So, we havewhere
As before, by eliminating the common factor , we obtain
Whence, it follows that
While handling (117), we decompose it as
In the series expression of we factor out as before and eliminate the common factor to obtain
Finally, substituting the expressions for and , we have
This finishes the proof of Theorem 7 (62).
12. Proof of Corollary 1
We conclude this chapter by finally proving Corollary 1. We add (114) and (124) and derive that
Now, in the above expression, on the left hand side and secant zeta function on the right hand side will cancel each other, as they are the same (from (50)). Therefore, we haveand thus Corollary 1 follows.
13. Future Work
By the virtue of the work of Lewittes, Berndt, and Arakawa, it would be interesting to find the general modular transformation formula for for all and from which one would like to see the truth of our Conjecture 2.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
B. Maji wishes to thank the Indian Institute of Technology Indore for the conductive environment.