Porter's constant

In mathematics, Porter's constant C arises in the study of the efficiency of the Euclidean algorithm.^[1][2] It is named after J. W. Porter of University College, Cardiff.

Euclid's algorithm finds the greatest common divisor of two positive integers m and n. Hans Heilbronn proved that the average number of iterations of Euclid's algorithm, for fixed n and averaged over all choices of relatively prime integers m < n, is

${\displaystyle {\frac {12\ln 2}{\pi ^{2}\ln n+o(\ln n).}$

Porter showed that the error term in this estimate is a constant, plus a polynomially-small correction, and Donald Knuth evaluated this constant to high accuracy. It is:

${\displaystyle {\begin{aligned}C&={6\ln 2} \over {\pi ^{2}\left[3\ln 2+4\gamma -{24} \over {\pi ^{2}\zeta '(2)-2\right]-{1} \over {2}\\[6pt]&={6\ln 2}((48\ln A)-(\ln 2)-(4\ln \pi )-2)} \over {\pi ^{2}-{1} \over {2}\\[6pt]&=1.4670780794\ldots \end{aligned}$

where

${\displaystyle \gamma }$ is the Euler–Mascheroni constant

${\displaystyle \zeta }$ is the Riemann zeta function

${\displaystyle A}$ is the Glaisher–Kinkelin constant

(sequence A086237 in the OEIS)

${\displaystyle -\zeta ^{\prime }(2)={\pi ^{2} \over 6}\left[12\ln A-\gamma -\ln(2\pi )\right]=\sum _{k=2}^{\infty }{\ln k} \over {k^{2}$

${\displaystyle }$

• Lochs' theorem
• Lévy's constant