SERIES EXPANSION OF A COTANGENT SUM RELATED TO THE ESTERMANN ZETA FUNCTION

In this paper, we study the cotangent sum c0 ( q p ) related to the Estermann zeta function for the special case when the numerator is equal to 1 and get two useful series expansions of c0 ( 1 p ) .


Introduction
For a positive integer p and q = 1, 2, . . . , p−1, such that (p, q) = 1, let the cotangent sum (see [10]) c 0 q p is the value at s = 0, It is well-known that the sum c 0 q p satisfies the reciprocity formula (see [2]) The Vasyunin cotangent sum (see [11]) arises in the study of the Riemann zeta function by virtue of the formula (see [2,9]) This formula is connected to the approach of Nyman, Beurling and Báez-Duarte to the Riemann hypothesis (see [8]), which states that the Riemann hypothesis is true if and only if lim

and the infimum is taken over all Dirichlet polynomials
In a recent work with A. Bayad [7], we have proved that the sum V q p satisfies the reciprocity formula Thereafter the restriction of the relationship (1.1) to q = 1 gives Exactly our interest in this work is the case q = 1 in order to get two series expansions of c 0 In [9, Theorem 1.7] H. Maier and M. Th. Rassias provide the following improvement. Let b, n ∈ N, b ≥ 6N , with N = n 2 +1. There exist absolute real constants A 1 , A 2 ≥ 1 and absolute real constants E l , l, with |E l | ≤ (A 1 l) 2l , such that for each n ∈ N we have Only in [9, Theorem 1.9] H. Maier and M. Th. Rassias provide another improvement, We draw attention that S. Bettin finds other reformulations of c 0 1 p inspired from continued fraction theory (see [3]).
Finally from another point of view we show in [5] with A. Bayad and M. O. Hernane that There is a misprint in the formula (1.22) Corollary 1.2 in [5] the correct one is in the formula (1.21) Corollary 1.2. Otherwise in the same paper [5], an integral representation of c 0 ( 1 p ) is given by In this work we prove that and we get another formulation that is Applying some techniques from the generating function theory [4] to previous integrals; we find two series expansions of c 0 1 p , as they are well explained in the next section.
2. Series Expansion of c 0 1 p Let b k be the integer sequence defined by b 0 = 1, b 1 = 2 and the recursive formulae: According to the terms b k we get the first series expansion in the following theorem.
For p ≥ 1 we define the arithmetic function a p in the form This function is not multiplicative. In general the arithmetical functions are defined from the set of natural integers N into C. We can extend this definition to F (C, C); set of functions from C to C. In that case the corresponding function is A : where H k is the Harmonic number Following this function a second series expansion of c 0 1 p is given in the following theorem.
2.1. Proof of Theorem 2.1. We take inspiration from the theory of generating functions [4,6], and prove that the sequence (b k ) is generated by the rational function: More precisely we get the following lemma. Lemma 2.1.
Proof. It is well known that is developable on entire series to get the result we have to take the quantity 2x − x 2 + x p − 2x p+1 + x p+2 instead of x in the last formula (2.4). Now, To compute this we use the well known Cauchy product of two entire series which generates the product of a polynomial of degree n with an entire series that also gives an entire series as follows We return to f (x) in writing with a 0 = 1, a 1 = −2, a 2 = 1, a p = −1, a p+1 = 2, a p+2 = −1, and the others are zero. We conclude that d 0 = 1, Finally, we see that d k and b k are identical for every integer k ≥ 0. For more information on this approach we refer to [6].
To get the result (2.1) of Theorem 2.1 we must substitute the expression (2.3) in the identity (1.2) and one obtains Furthermore, Finally, and c 0 (1) = c 0 1 2 = 0 is compatible with the definition of c 0 . Regarding the identity (2.3) Lemma 2.1 we remark that Furthermore, for x = 1 2 we deduce that the coefficients b k satisfy the following statements (2.5) Proof.