Criteria for irrationality of Euler’s constant

By Jonathan Sondow. 2002

60 citations recorded by the source corpus.

By modifying Beukers' proof of Apery's theorem that ζ(3) is irrational, we derive criteria for irrationality of Euler's constant, γ. For n > 0, we define a double integral In and a positive integer S n , and prove that with d n = LCM(1,..., n) the following are equivalent: 1. The fractional part of log S n is given by {log S n } = d 2n I n for some n. 2. The formula holds for all sufficiently large n. 3. Euler's constant is a rational number. A corollary is that if {log S n } > 2 -n infinitely often, then γ is irrational. Indeed, if the inequality holds for a given n (we present numerical evidence for 1 < n < 2500) and γ is rational, then its denominator does not divide d 2n ( 2n n). We prove a new combinatorial identity in order to show that a certain linear form in logarithms is in fact log S n . A by-product is a rapidly converging asymptotic formula for γ, used by P. Sebah to compute γ correct to 18063 decimals.

Subfields: Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/f52f052192b8ef34c6636af26e9a86c0d1957484

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In Mathematical Physics

1. Michel Waldschmidt
2. Jeffrey C. Lagarias
3. Jesús Guillera
4. Jonathan Sondow
5. Jonathan Pila
6. Alex Wilkie

In Numerical Analysis

1. Michel Waldschmidt
2. Jeffrey C. Lagarias
3. Jesús Guillera

Citations in the corpus

• The generalized-Euler-constant function γ(z) and a generalization of Somos's quadratic recurrence constant (2006, unknown): “In [15, 16, 17], the first author represented Euler’s constant γ = γ(+1) and the alternating Euler constant log 4 π = γ(−1) by the double integrals γ(±1) = ∫ 1 0 ∫ 1 0 1− x (1∓ xy)(− log xy) dx dy.”
• Open Diophantine problems (2003, background): “IV) proved independently in 1936 that g(k) = I(k) provided that the number r = 3k − 2kq satisfies r ≤ 2k − q − 2.”
• Euler's constant: Euler's work and modern developments (2013, background): “Sondow [276], [278] and Sondow and Zudilin [282].”
• Transcendence of Periods: The State of the Art (2006, background): “(see [56]), even when a stronger assertion is expected, namely:”
• Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent (2005, background): “is Euler’s constant, we recover the formula in [20], [22], and [23]”