Sondow, Jonathan Double integrals for Euler’s constant and \(\ln \frac 4\pi\) and an analog of Hadjicostas’s formula. (English) Zbl 1138.11356 Am. Math. Mon. 112, No. 1, 61-65 (2005). From the introduction: Euler’s constant \(\gamma\) is defined as the limit\[ \gamma= \lim_{N\to\infty} \bigl(1+ \tfrac12+ \tfrac13+\cdots+ \tfrac1N-\ln N\bigr). \tag{1} \]In this note we prove the formulas\[ \gamma= \sum_{n=1}^\infty \biggl(\frac1n-\ln \frac{n+1}{n}\biggr)= \iint_{[0,1]^2} \frac{1-x}{(1-xy)(-\ln xy)}\,dx\,dy, \tag{2} \] \[ \ln \frac 4\pi= \sum_{n=1}^\infty (-1)^{n-1}\biggl(\frac1n-\ln \frac{n+1}{n}\biggr)= \iint_{[0,1]^2} \frac{1-x}{(1+xy)(-\ln xy)}\,dx\,dy. \tag{3} \]In view of series (2), which is due to Euler, series (3) reveals \(\ln(4/\pi)\) to be an “alternating Euler constant”. Cited in 2 ReviewsCited in 27 Documents MSC: 11Y60 Evaluation of number-theoretic constants PDFBibTeX XMLCite \textit{J. Sondow}, Am. Math. Mon. 112, No. 1, 61--65 (2005; Zbl 1138.11356) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Decimal expansion of Euler’s constant (or the Euler-Mascheroni constant), gamma. (Number of 0’s) - (number of 1’s) in the base-2 representation of n. Decimal expansion of the ”alternating Euler constant” log(4/Pi). Continued fraction for the ”alternating Euler constant” log(4/Pi). Numerators in a series for the ”alternating Euler constant” log(4/Pi).