Properties and Applications of the Reciprocal Logarithm Numbers

Abstract

Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the reciprocal of the logarithmic function ln(1 + z), whose coefficients represent an infinite set of fractions. These numbers, which are called reciprocal logarithm numbers and are denoted by A k , converge to zero as k → ∞. Several properties of the numbers are then obtained including recursion relations and their relationship with the Stirling numbers of the first kind. Also appearing here are several applications including a new representation for Euler's constant known as Hurst's formula and another for the logarithmic integral. From the properties of the A k it is found that a term of ζ(2) cannot be eliminated by the remaining terms in Hurst's formula, thereby indicating that Euler's constant is irrational. Finally, another power series expansion for the reciprocal of arctangent is developed by adapting the preceding material.

Erratum to “A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations” [J. Number Theory 148 (2015) 537–592]

“…a k are coefficients in the Maclaurin expansion of z/ ln(1 + z) and are usually referred as to (reciprocal) logarithmic numbers or Gregory's coefficients (in particular a 1 = 6 Fontana-Mascheroni's series (7) seems to be the first known series representation for Euler's constant containing rational coefficients only and was subsequently rediscovered several times, in particular by Kluyver in 1924 [52], by Kenter in 1999 [51] and by Kowalenko in 2008 [55] (this list is far from exhaustive, see e.g. [56]).…”

“…We already corrected many of them in our previous work [10, Sections 2.1 & 2.3]. As regards the above-referenced Malmsten's original equation (55), case m + n even, note that Γ( n−i 2n ) should be replaced by Γ( n−i n ). Formula (56) also has an error: Γ( n+i n ) should be replaced by Γ( n−i n ).…”

Generalizing the Reciprocal Logarithm Numbers by Adapting the Partition Method for a Power Series Expansion

“…In a recent work [1] a relatively novel number theoretic/graphical method was used to derive a new power series expansion for the reciprocal of the logarithmic function, viz. z/ ln(1 + z).…”

“…Because of this similarity it was intimated that the method used in Ref. [1] could be adapted to calculate the Bernoulli numbers despite the fact that they diverge as k → ∞. In this work we aim to describe how this method, which shall be referred to as the partition method for a power series expansion, can be adapted to obtain the Bernoulli numbers.…”

“…In deriving this power series expansion, we will be generalizing the reciprocal logarithm numbers presented in Ref. [1]. This paper is arranged as follows.…”