Some Definite Integrals Associated with the Riemann Zeta Function

The authors aim at deriving a family of series representations for ζ(2n+1) (n ∈ N) by evaluating certain trigonometric integrals in several different ways. They also show how the results presented in this paper relate to those that were obtained in other works. Finally, some illustrative computational examples, using Mathematica (Version 4.0) for Linux, are considered.


Introduction
The Riemann Zeta function ζ(s) and the Hurwitz (generalized) Zeta function ζ(s, a), which are defined usually by can indeed be extended meromorphically to the whole complex s-plane except for a simple pole at s = 1 with residue 1 (see, for details, Titchmarsh [23]), N 0 being the set of nonnegative integers. It is easily seen from (1.1) and (1. In recent years there has been a renewed interest in representing ζ(2n + 1) (n ∈ N), N being the set of positive integers, by means of series which converge more rapidly than the defining series in (1.1). These developments seem to have stemmed from the use of the familiar series representation (see, e.g., Hjortnaes [14] and Gosper [11]): in Apéry's proof [2] of the irrationality of ζ (3), as well as from Ewell's yet another rediscovery [8] of Euler's formula (see, e.g., Ayoub [3: pp. 1084Ayoub [3: pp. -1085; see also Ramaswami [19] and Srivastava [20: p. 7/Equation (2.23)]): (1.5) The works of (among others) Ewell [9], Dabrowski [7], Zhang and Williams [24], Cvijović and Klinowski [6], and Srivastava [21,22] may be cited in connection with the aforementioned developments.
The main object of this paper is to derive a family of series representations for ζ(2n + 1) (n ∈ N) by evaluating certain trigonometric integrals in several different ways and to show how the results presented here relate to those that were obtained in other works. We also consider some illustrative computational examples by using Mathematica (Version 4.0) for Linux.

A cosecant integral and its consequences
We begin by considering the cosecant integral which, upon integration by parts, readily yields we find from (2.2) that The Riemann Zeta Function 833 for (s) > 1 and ω > 1. In its special case when ω = 2, the integral formula (2.4) reduces immediately to the form: which, for s ∈ N \ {1}, appears in the works of (for example) Glasser  The integral formula (2.4) also simplifies when ω = 3 and ω = 6, giving us the special cases for (s) > 1, which do not seem to have been recorded earlier.

Series representations for ζ(2n + 1)
In every situation in which the Clausen functions C 2n and C 2n+1 can be expressed in closed forms, each of our formulas (2.18) and (2.19) will readily yield a series representation for ζ(2n + 1) (n ∈ N). We begin by considering the following rather simple special case.
Thus the formulas (2.18) and (2.19) yield the series representations for n ∈ N, respectively. In its special case when n = 1, the result (3.2) immediately reduces to the following series representation for ζ(3): (3.5) The Riemann Zeta Function

837
In view of the known sum: which incidentally results also from (2.18) in the special case when ω = 2 and n = 1, Euler's formula (1.5) is indeed a simple consequence of (3.5).
We remark in passing that an integral representation for ζ(2n + 1), which is easily seen to be equivalent to the series representation (3.3), was given by Dabrowski [7: p. 203/Equation (16) The generalized Clausen function C 2n+1 defined in (2.17) is known to be expressible in a closed form in at least three other cases, and we find for n ∈ N that (cf. Lewin [16: p. 198 With a view to evaluating the generalized Clausen function C 2n , also defined in (2.17), when x = 1 2 π, x = 1 3 π and x = 2 3 π, we recall the following result recorded (for example) by Hansen  for (s) > 1 and 0 < x < 2π. By setting x = 2π ω (ω > 1) and y = 0, (3.10) reduces at once to the form: for (s) > 1 and ω > 1. Now, by making use of the identity: for p, q ∈ N, as well as the familiar special case of (3.13) when p = q = 1 or, equivalently, we find from (3.11) and the relevant definition in (2.17) that, for n ∈ N, (3.18) in which each ζ(2n) can be replaced by its value in terms of the Bernoulli numbers B 2n by appealing to the well-known relationship (cf., e.g., [17: p. 19]): Much more general trigonometric sums than the ones involved in (3.7) to (3.9) and (3.16) to (3.18) can also be evaluated by applying the aforementioned technique or (alternatively) by appealing appropriately to the elementary series identity: and its straightforward consequence (3.12). Thus, for example, we can derive a generalization of (3.7) in the form: (3.21) which immediately yields (3.7) in the special case when s = 2n + 1 (n ∈ N). Analogous generalizations of the remaining evaluations of the Clausen functions C 2n+1 and C 2n can be derived similarly; we choose to skip the details involved.
In view of the evaluations (3.7) to (3.9) and (3.16) to (3.18), we are led easily to the following additional special cases of our general results (2.18) and (2.19).
Case 2. Let ω = 3. Then, by making use of (3.9) and (3.18) in each of our formulas (2.18) and (2.19), we obtain the series representations and ζ(2n for n ∈ N. In particular, when n = 1, (3.22) and (3.23) yield the following (presumably new) series representations for ζ(3): and Case 3. Let ω = 4. Then, by applying the evaluations (3.7) and (3.16) in each of our formulas (2.18) and (2.19), we obtain the series representations ζ(2n + 1) = (−1) n−1 (2π) 2n (2n + 1)!(2 2n − 1) log 2 + 4 ∞ k=0 ζ(2k) (2k + 2n + 1)4 2k and for n ∈ N. In their special cases when n = 1, (3.26) and (3.27) yield the following (presumably new) series representations for ζ(3): The Riemann Zeta Function 841 and ζ(2n + 1) for n ∈ N. In their special cases when n = 1, these last results (3.30) and (3.31) yield the following (presumably new) series representations for ζ (3): . (3.33) 4. An alternative derivation of the series representation (3.2) In view of the following known representation of ζ(s) as a Mellin transform [17: p. 21]: we propose to give here an interesting alternative derivation of the very first of the set of eight series representations for ζ(2n + 1) (n ∈ N) which we have presented in the preceding section. Indeed, from the work of Glasser [10: p. 445/Equation (4) with α → 0], it is known also that for n ∈ N. Now evaluate the cosecant integral in (4.2) by means of the known result (2.5) with s = 2n (n ∈ N). Thus, by setting log u = −2t, we find from the last member of (4.2) that for n ∈ N. By the binomial expansion, we find for n ∈ N 0 that where, as also elsewhere in this work, an empty sum is interpreted to be nil. Upon replacing n in (4.3) by n + 1 and making use of (4.4), we obtain, for n ∈ N 0 , The infinite integral in (4.5) can be evaluated by means of the known result (4.1) except when j = 0, in which case it is easily seen by integrating by parts that (cf. for n ∈ N 0 . For n = 0, (4.7) immediately yields the known sum (3.6), which (as we observed in the proceding section) is derivable also from (2.18) for ω = 2 and n = 1. More interestingly, just as we indicated in Section 3, since 1 (2k + 1)(2k + 2) There is yet another approach for the derivation of a closed-form expression for the infinite series in (5.11). This general (generating-function) approach is based upon a familiar integral representation for the Zeta function; it was fully described and illustrated by Adamchik and Srivastava [1], whose technique is also implemented in Mathematica (Version 3.0). Here we choose to present three examples which demonstrate how Mathematica (Version 4.0) for Linux can be used to evaluate the infinite series in (5.11) for certain values of the parameters p and ω: Since ζ(0) = − 1 2 , these examples completely agree with the special cases (3.6), (3.5), and (3.4), respectively, which have already been shown here to follow from our general series representations considered in Section 3.