Proceedings of the World Congress on Engineering 2014 Vol II, WCE 2014, July 2 - 4, 2014, London, U.K. Euler’s Constant: New Insights by Means of Minus One Factorial Okoh Ufuoma Abstract—The great object of this paper is to furnish, in German mathematician Johann Carl Friedrich Gauss (1777– a concise and plain manner, new insights into that mysterious 1855), evaluated γ up to 40 correct decimal places in accor- constant whose arithmetic nature was shrouded in obscurity dance with Soldner’s value [6], [15], [32]. for over 250 years, the famous Euler’s constant. The cardinal The constant whose numerical value is γ = instrument which generates these new insights is the unwonted 0:577215664 ::: [14], [23], a real number, forms a triple infinite number, the minus one factorial, which in this paper is constant with two famous and important real constants, discussed in detail. π = 3:141592653 ::: and e = 2:718281828 ::: as it is often ranked the third most important real constant after π and e Index Terms—Euler’s constant, infinity, harmonic number, [6], [14], not based on its position in respect of magnitude, harmonic series, natural logarithm, minus one factorial for example, π > e > γ, but depending on its position with regard to its ubiquity in mathematics, revealed by its frequent I. INTRODUCTION appearance in equations. Of the triple constant it is only γ NE of the most enchanting mathematical discoveries of whose arithmetic nature was an enigma to mathematicians O the 18th century was the mysterious and celebrated as its irrationality and transcendence were unfathomable for constant, the Euler’s constant γ, discovered by the immor- many years [14] . The irrationality of π and e was proved by a tal Swiss professor in mathematics in Berlin and St Peters- Swiss mathematician Johann Heinrich Lambert (1728–1777) burg, Leonhard Euler (1707-1783) [23], who, in his ad- in 1761 and that of eπ by a Soviet mathematician Alexander mirable paper titled De progressionibus harmonics observa- Osipovich Gelfond (1906–1968) in 1929 [16]. tiones (1734/5) [10], defined the constant in a fascinating The main aim of this paper is to provide new insights into manner as the character of γ by means of the minus one factorial (−1)! 1 1 1 1 together with other relevant results which we shall also con- γ = lim 1 + + + + ··· + − ln(n + 1) n!1 2 3 4 n sider in this paper. We shall here employ (−1)! to derive, in a systematic manner, a new and beautiful expression for γ. and computed its numerical value to 6 decimal places as The rest of the paper is structured into seven sections. Sec- tion II provides, for the reader to evince enthusiastic interest γ = 0:577218: in the paper, a brief review of Euler’s quest for the arithmetic In the 1st section of another paper entirely devoted to the in- nature of γ. Section III introduces the new and unfamiliar vestigation of γ and intriguingly entitled De numero memo- constant (−1)! while Section IV elegantly treats its arithmetic rabili in summatione progressionis harmonicae naturalis oc- nature. Section V discusses precise details of the harmonic currente [11] [20], Euler painstakingly computed the value of number and the natural logarithm, the two concomitants re- the constant to 16 decimal places as sponsible for the creation of the Euler’s constant. Section VI introduces for the first time the natural logarithm of (−1)! γ = 0:5772156649015325: as the sum of the divergent harmonic series. That the reader may see the great beauty and force of the new constant (−1)!, The constant is sometimes called Euler–Mascheroni Section VII treats the use of the natural logarithm of (−1)! for constant [22] to honour together with Euler the Italian deriving a familiar formula that links the Euler’s constant with mathematician Lorenzo Mascheroni (1750–1800) [6], prob- the famous Riemann zeta constants. Finally, Section VIII pro- ably the second to intensively investigate the constant. In vides for the first time a new and interesting identity that for- Adnotationes ad calculum 1790 Mascheroni published his ever links γ with (−1)!. integralem Euleri [6], [14] and there computed γ up to 32 decimal places, denoting it with the letter A. A few years II. EULER’S QUEST FOR THE ARITHMETIC later, in 1809, a German mathematician, Johann Georg von NATURE OF γ Soldner (1766–1833) found a value of the constant which The discovery of the Euler’s constant, γ, originated from a was in harmony with only the first 19 decimal places of famous problem, the Basel problem, posed in the 17th century Mascheroni’s computed value. It was in 1812 that a young in a 1650 book Novae Quadraturea Arithmeticae by a profes- prodigious mathematician Friedrich Bernhard Gottfried sor of mechanics in Bologna, Pietro Mengoli (1625 − 1686), Nicolai (1793–1846), under the supervision of the famous an Italian mathematician and clergyman [1]. The Basel prob- lem asked for the exact sum of the series 1 Manuscript received Feb. 12, 2014; revised March 1, 2014. Okoh X 1 1 1 1 Ufuoma is with the Department of Mathematics/Science, Apex Academy, = lim 1 + + + ··· + n2 n!1 22 32 n2 Uwheru, Delta State, Nigeria (tel: 08134813234; e-mail: okohu- n=1 [email protected], [email protected]) together with a proof that the sum was precise [1], [14]. ISBN: 978-988-19253-5-0 WCE 2014 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2014 Vol II, WCE The2014, eminent July 2 - English 4, 2014, mathematician London, U.K. and professor at Ox- tain the following tabular form: ford John Wallis (1616–1703) had communicated in his 1655 1 1 1 1 book Arithmetica Infinitorum that he had got the sum to three 1 = ln (2) + − + − + ··· decimal places as 1:645, but he was unable to say anything 2 3 4 5 1 3 1 1 1 1 more concrete [1]. The problem became notable because it = ln + − + − + ··· was also raised by one of the eight prominent mathematicians 2 2 2 · 4 3 · 8 4 · 16 5 · 32 produced by a remarkable Swiss family in three generations, 1 4 1 1 1 1 = ln + − + − + ··· Jakob Bernoulli (1654 − 1705), a professor of mathematics 3 3 2 · 9 3 · 27 4 · 81 5 · 243 at the University in Basel. He attempted the problem with- . out success, and in his posthumous book, the illustrious Ars . Conjectandi, published in 1713, he appealed [4], [20]: 1 n + 1 1 1 1 1 = ln + − + − + ··· : n n 2 · n2 3 · n3 4 · n4 5 · n5 ::: it is more difficult than one would have ex- pected, which is noteworthy. If someone should Adding by columns the first n terms, we get succeed in finding what till now withstood our ef- 1 1 1 forts and communicate it to us, we would be much 1 + + + ··· + = ln(n + 1) 2 3 n obliged to them. 1 1 1 1 + 1 + + + ··· + 2 4 9 n2 This problem prompted Euler’s research and he spent 1 1 1 1 much effort evaluating the sum of reciprocals of powers − 1 + + + ··· + 3 which in modern notation is written as 3 8 27 n 1 1 1 1 + 1 + + + ··· + 1 4 16 81 n4 X 1 ζ(k) = ; k ≥ 1 1 1 1 1 nk − 1 + + + ··· + n=1 5 32 243 n5 + ··· : where ζ(k) is a zeta constant. In 1735 he obtained the exact sum 1 1 If n tends to the infinitely large integer Ω so that 1 + 2 + 3 + ··· + 1 becomes the harmonic series, we have π2 n ζ(2) = 6 1 1 1 1 + + + ··· + = ln(Ω + 1) + γ 2 3 Ω and announced his noteworthy solution to the Basel problem where [1], [14]. He also procured the exact values for higher even zeta constants ζ(2m), but did not succeed in finding the exact 1 1 1 1 γ = 1 + + + ··· + value for any odd zeta constant ζ(2m − 1). In a letter to his 2 4 9 Ω2 tutor, the notable Johann Bernoulli (1667– 1748), the younger 1 1 1 1 brother of Jakob, he wrote, “::: the odd powers I cannot sum, − 1 + + + ··· + 3 8 27 Ω3 and I don’t believe that their sums depend on the quadrature 1 1 1 1 of the circle [π] ”[14]. + 1 + + + ··· + In an attempt to assign a value to the odd zeta constant 4 16 81 Ω4 ζ(1), also called the harmonic series, he discovered his fa- 1 1 1 1 − 1 + + + ··· + vorite constant which we now call the Euler’s constant γ, 5 32 243 Ω5 computed it to six decimal places and published his results in + ··· the paper [10], and there denoted his constant with the letter 1 1 1 1 C, stating that it was “worthy of serious consideration”[17]. = ζ(2) − ζ(3) + ζ(4) − ζ(5) + ··· Now the birth of Euler’s constant as Euler presented it in 2 3 4 5 1 π2 1 1 π4 his paper [10] is as follows: We start with the Taylor series = − (1:202056903 :::) + − · · · expansion of ln(1 + x), 2 6 3 4 90 = 0:577215664 :::: x2 x3 x4 x5 ln(1 + x) = x − + − + − · · · ; 2 3 4 5 Euler, as was his custom, would never give up the search for anything he thought deserves serious consideration. He which was discovered by a 17th-century German mathemati- engaged himself with great ardor in a long search for the arith- cian Nicholas Mercator (c.1620–1687). We rearrange this se- metic nature of γ, desiring to know the sort of number it is, ries as whether or not it is irrational, incapable of being written as a ratio or quotient of two integers, and transcendental, inca- pable of being expressed as a polynomial.