Criteria for irrationality of Euler’s constant

2003/03/11 by Jonathan Sondow · 4 citations
Mathematics · #Mathematical and Theoretical Analysis #Iterative Methods for Nonlinear Equations #Irrationality #Constant (computer programming) #Euler's formula #Mathematics #Mathematical economics #Mathematical analysis #Computer science #Rationality #Philosophy #Epistemology #Programming language

paper · pdf · doi:10.1090/s0002-9939-03-07081-3

openalex publication_date 2003/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15

Abstract

By modifying Beukers’ proof of Apéry’s theorem that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta left-parenthesis 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi> ζ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">ζ (3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is irrational, we derive criteria for irrationality of Euler’s constant, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma"> <mml:semantics> <mml:mi> γ </mml:mi> <mml:annotation encoding="application/x-tex">γ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we define a double integral <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript n"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">I_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and prove that with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript n Baseline equals upper L upper C upper M left-parenthesis 1 comma ellipsis comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>LCM</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">d_n=\operatorname LCM(1,\dotsc ,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the following are equivalent: 1. The fractional part of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="log upper S Subscript n"> <mml:semantics> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">log S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is given by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace log upper S Subscript n Baseline right-brace equals d Subscript 2 n Baseline upper I Subscript n"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\log S_n\=d_2nI_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 2. The formula holds for all sufficiently large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 3. Euler’s constant is a rational number. A corollary is that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace log upper S Subscript n Baseline right-brace greater-than-or-equal-to 2 Superscript negative n"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mo> ≥ </mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> −

Citations

• Approximate formulas for some functions of prime numbers

Cited by

• A hypergeometric approach, via linear forms involving logarithms, to criteria for irrationality of Euler’s constant
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• Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent
• Euler's constant [wikipedia]