IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 15, Issue 1 Ser. I (Jan – Feb 2019), PP 57-70 www.iosrjournals.org DOI: 10.9790/5728-1501015770 www.iosrjournals.org 57 | Page Fluctuation around the Gamma function and a Conjecture Danilo Merlini 1 , Massimo Sala 2 and Nicoletta Sala 3 1 CERFIM/ISSI, Locarno, Switzerland 2 Independent Researcher 3 Institute for the Complexity Studies, Rome, Italy Abstract: Using the expansion of the log of the ξ and the ζ functions in terms of the Pochammer's Polynomials, we obtain a fast convergence sequence for the first two Li-Keiper coefficients. The sequences are of oscillatory type. Then we study the oscillating part of the Li-Keiper coefficient (the “tiny” oscillations) and following some analytical calculations we pose a new conjecture in the form of a kind of “stability bound” for the maximum strength of the fluctuations around the mean staircase. Key words: Pochammer’s Polynomials, ξ function (Xi), Li-Keiper coefficients, Baez-Duarte and Maslanka expansions, trend and “tiny” fluctuations. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 25-01-2019 Date of acceptance: 07-02-2019 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction Recently, Matiyasevich has given a new interesting Formula for the first Li coefficient, i.e. a representation of it with positive summands by means of the binary sequence for the Euler constant γ and for log(π), which reads [1]: λ 1 =∑( 1   + 1 1−  )= ∞ =1 1 2 ∙ ( − log(4) + 2) = ∑ ∞ =3 1 2 ∙ (2 1 () + 3) [(2) ∙ (2 + 1) ∙ (2 + 2)] Where: 1   is the reciprocal of any nontrivial zero of the Zeta function and N1(n) is the number of units i.e. of 1 in the binary expansion of n. See also previous related works [2, 3]. In the first part of this work we will find another representation of λ1 and of λ2 which is not in the form of positive summands but that it is given by an alternating sequence of rational numbers and Zeta value of half integer arguments emerging from the representation of the ξ function. The sequence is nevertheless “fast” converging to the true values of the two constants. We use a Pochammer's representation of ξ at special values of the parameters (α, β) given in a systematic analysis of some representation of the Zeta function [4, 5, 6, 7, 8], that is here α =1/2, β =1. We do not comment on a point of view present in the literature concerning the utility or not of a representation of the Zeta function involving values of the Zeta function at integer arguments [9, 10, 14]. Some serious treatments have appeared in these years and various numerical experiments have been extensively pursued in addition to some rigorous partial results obtained in some pioneering works [4, 5,12 ]. Here, since dealing with representations of functions at the border of the domain of absolute convergence (i.e. at s ~ 1 ) no additional proofs seems to be necessary and our strategy is motivated by the results of our analytical and numerical experiments. II. The expansion of the log of ξ (z) and others functions by means of Pochammer's Polynomials We start with the expression of ξ (s), i.e. the Xi function where s = σ + i· t is the usual complex variable. Introducing the new variable z given by z = 1-1/s i.e. s = 1/(1-z), so that the critical line s = 1/2 + i· t is mapped onto the unit circle abs( z) = |z| = 1 we then have [13, 14] ξ (z) = (½) · z· [1/(1-z) 2 ] · π -(1/(2 · (1-z)). Γ(1/(2· (1-z))) · ζ(1/(1-z)) (1) where Γ(s/2) is the Gamma function of argument s/2 and ζ(s) is the Zeta function of argument s. Notice also that π -(s/2) = π -1/(2 · (1-z)) . We then consider the Pochammer's Polynomials in the complex variable s of degree k , given by [4 ,5]: