2013/03/31 by Jeffrey C. Lagarias · 4 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #math.NT #msc:01A50 #msc:11J02 #msc:11J72 #msc:11J81 #msc:11M06
paper · pdf · doi:10.1090/s0273-0979-2013-01423-x
published as Bulletin Amer. Math. Soc. 50 (2013), No. 4, 527--628 · v4 98 pages (pagewidth decreased), 321 references, text agrees with published version (but with less reference information); v5 post-publication, introduces a mistake, Theorem 3.1.3 is ok as originally stated, v6, like the monkey's paw, changes back, but corrects typo in (3.1.8) and in ζ(1-k) formula two lines following
openalex publication_date 2013/07/19 · arxiv created 2013/10/25 · arxiv updated 2013/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper has two parts. The first part surveys Euler’s work on the constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma equals 0.57721 midline-horizontal-ellipsis"> <mml:semantics> <mml:mrow> <mml:mi> γ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.57721</mml:mn> <mml:mo> ⋯ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">γ =0.57721⋯</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant.