Euler’s integral, multiple cosine function and zeta values

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In this paper, as a generalization, we evaluate the definite integrals x 0 θ r−2 log cos θ 2 dθ for r = 2, 3, 4, . . . .We show that it can be expressed by the special values of Kurokawa and Koyama's multiple cosine functions C r (x) or by the special values of alternating zeta and Dirichlet lambda functions.
In particular, we get the following explicit expression of the zeta value where G is Catalan's constant and C 3 1 4 is the special value of Kurokawa and Koyama's multiple cosine function C 3 (x) at 1  4 .Furthermore, we prove several series representations for the logarithm of multiple cosine functions log C r The main purpose of this paper is to relate the Euler type integrals and the multiple cosine functions with the special values of zeta functions.So in this section, to our purpose, firstly we introduce various types of zeta functions.For Re(s) > 1, the Riemann zeta function is defined by This function can be analytically continued to a meromorphic function in the complex plane except for a simple pole, with residue 1, at the point s = 1.The special number ζ(3) = 1.20205 • • • is called Apéry constant.It is named after Apéry, who proved in 1979 that ζ(3) is irrational (see [4]).For Re(s) > 1 and a = 0, −1, −2, . . ., in 1882, Hurwitz [14] defined the partial zeta function (1.2) ζ(s, a) = ∞ n=0 1 (n + a) s , which generalized (1.1).As (1.1), this function can also be analytically continued to a meromorphic function in the complex plane except for a simple pole at s = 1 with residue 1.
The Dirichlet lambda function λ(s) is defined by (1.9) for Re(s) > 1 (see [13, p. 954, (1.9)]).This function was studied by Euler under the notation N(s) (see [34, p. 70]).Euler also considered its alternating form for Re(s) > 0, which he denoted by L(s) (see [34, p. 70]).Furthermore, the constant β(2) = G is usually named as Catalan's constant (see [21], [30], [32] and [36]).Both functions admit an analytic continuation, λ(s) to all s = 1 and β(s) to all s.They have been studied in detail by us in [13], in particular, we have obtained a number of infinite families of linear recurrence relations for λ(s) at positive even integer arguments λ(2m), convolution identities for special values of λ(s) at even arguments and special values of β(s) at odd arguments.
The Dirichlet L-function associated to a Dirichlet character χ is given by which is convergent for Re(s) > 1 and the Euler product is taken over all prime numbers p.It was introduced by Dirichlet in 1837 to prove the theorem on primes in arithmetic progressions (see [5,Chapter 7]).For the trivial Dirichlet character ½ we have L(s, ½) = ζ(s).For the principal character ½ m of modulus m induced by ½ we have [15, p. 255] (1.12) We may also express L(s, χ) by using the Hurwitz zeta functions.Let f be a positive integer and let χ be any character modulo f.The Dirichlet L-function L(s, χ) is expressed in terms of the Hurwitz zeta function ζ(s, a) by means of the following formula for Re(s) > 1, and it can be analytic continued to the whole s-plane from the above expression.

Multiple trigonometric functions and the related integrals.
Around 1742, Euler successfully calculated the zeta value In concrete, by taking x = 1 in the left-hand side we get π 2 8 , and by expanding arcsin t as a power series and integrating term-by-term on the right-hand side, we get the sum , where λ(s) is the Dirichlet lambda function (see (1.9)).Then by comparing the results on the both sides we arrive at the summation Finally, the identity (1.17) And more generally, for n = 1, 2, 3, . . ., Euler obtained where the B 2n are the Bernoulli numbers defined by the generating function (See [5, p. 266, Theorem 12.17]).But the explicit formulas for ζ(3) and ζ(2n + 1) are still unknown.For the long-standing history, we refer to a recent book by Nahin [29].

Main results
In this section, we state our main results.Their proofs will be given in Section 4. First, we represent (1.35) by the special values of multiple cosine functions.
In the following, we shall employ the usual convention that an empty sum is taken to be zero.For example, if n = 0, then we understand that n k=1 = 0. We represent (1.35) with x = π 2 by the special values of alternating zeta, lambda and beta functions.Now we state the following result.
Theorem 2.4.For r = 2, 3, 4, . . ., is one of famous mysterious constants appearing in many places in mathematics and physics.It can be represented by the special values of Hurwitz zeta functions (see [21, p. 667 Setting r = 2, 3, 4 and 5 in Theorem 2.4, by (2.1) we get the following corollary.
Corollary 2.13.For 0 ≤ x < 1, we have Remark 2.14.In particular, setting x = 1 2 in the above relations and by using the expansions we recover Corollary 2.6.

Multiple cosine functions
In this section, to our purpose, we state some basic properties of multiple cosine functions.Some of them have been reported in [22, p.  First, we prove the following proposition which is necessary to derive several properties of multiple cosine functions.Note that it has appeared in [22, p. 848] and [23, p. 124] without proof.Proposition 3.1.For r = 2, 3, 4, . . ., we have C r (0) = 1 and C r (x) is a meromorphic function in x ∈ C satisfying Thus we have the integral representation where the contour lies in C \ {± 1 2 , ± 3 2 , . ..}. Proof.The proof goes similarly with [17, Proposition 1] by calculating the logarithmic derivative.When r = 1, the result follows immediately from (1.31).For r = 2, 3, 4, . . ., by using and (1.29), we obtain Then by observing the expressions we see that where we have used the expansion [11, p. 43, 1.421 This completes the proof of Proposition 3.1.
From Proposition 3.1, we get a new proof for the following result by Kurokawa and Wakayama [23, p. 125].
Proposition 3.5.For r = 2, 3, 4, . . ., the multiple cosine function C r (x) satisfies the following second order algebraic differential equation On the other hand, by applying the derivative formula in calculus directly, we have (3.9)d dx Then by comparing (3.8) and (3.9), we get which is equivalent to the statement of the proposition.
Remark 3.6.Propositions 3.5 is an analogy of Painlevé's differential equation of type III.

Proofs of the results
In this section, we prove the results stated in Section 2.
The first proof of Theorem 2.1.As remarked by Allouche in an email to us, this result can be implied by (1.25) if using and noticing the relation between C r (x) and S r (x) (see (1.34)).Following his idea, we give a detailed proof as follows.From (4.1) and (1.25) we have (4.2) On the other hand, the logarithmic of (1.34) yields Then substituting (4.1) and (4.3) into (4.2),we get our result.
The second proof of Theorem 2.1.Here we also derive this result directly.From Proposition 3.3 and the integration by parts, we have Then changing the variable to θ = πt in this integral, we have Now, letting x → x π , the assertion follows.
Proof of Theorem 2.3.To prove this, we need the following two lemmas.
Lemma 4.2.For r = 0, 1, 2, . . ., we have , Proof.Setting x = π 2 in Lemma 4.1, by the fundamental formula of angle addition for the sine function, we obtain (4.4) For the calculation of the right hand side, we split the summation into three parts I 1 , I First we calculate the sum I 1 .From (1.10), we have (4.5) Then we calculate the sum I 2 .From (1.6), we have (4.6) Finally, (1.6) also implies that (4.7) Substituting (4.5), (4.6) and (4.7) into (4.4)we get the lemma.Now we go to the proof of Theorem 2.3.From the following series expansion (see [33, p. 148 for r = 2, 3, 4, . . . .Then replacing r by r − 2 in Lemma 4.2 and substituting the result into the right hand side of the above equation, after some elementary calculations, we obtain the desired result. Proof of Theorem 2.8.From Euler's infinite product representation of the cosine function (1.30), we have for |θ| < π 2 .In the above equation, replacing θ by θ 2 , then multiplying both sides by θ r−2 and integrating the result from 0 to x, we have (4.8) where 0 ≤ x < π.On the other hand, by Theorem 2.1 we have (4.9) Comparing (4.8) and (4.9) we get The result is now easily established.
(3) and ( 4) From (4.10) we have Then by taking the exponential on the both sides of the above equation, we get (4.11) Finally, by taking the real part in the above expression for 2 ≤ r ∈ 2Z and 3 ≤ r ∈ 1 + 2Z repectively, also notice that (see we obtain (3) and (4).

Miscellaneous results
In this section, we present several new representations for log C r (x) and some series involving λ(2k), the special values of Dirichlet's lambda function at positive even integer arguments.

x 2 by 1 . 1 .
zeta functions, L-functions or polylogarithms.One of them leads to another expression of ζ(3Zeta functions.
2 , I 3 according to the terms x = 1 4 in (5.22), and using Corollary 2.6 for r = 2, 3, 4 and 5, it is readily to obtain