Porter's constant

In mathematics, Porter's constantC 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 )

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

${\displaystyle }$

Approximations

The simple approximations of the Porter's constant accurate up to 3 digits can be found as

${\displaystyle C=2^{21/38}=1.466758730139474...,}$

accurate up to 4 digits as

${\displaystyle C=2^{26/47}=1.467328090156686...,}$

and up to 6 digits as

${\displaystyle C=2^{47/85}=1.467073525425601....}$

See also

• Lochs' theorem
• Lévy's constant

References

1. Knuth, Donald E. (1976), "Evaluation of Porter's constant", Computers & Mathematics with Applications, 2 (2): 137–139, doi:10.1016/0898-1221(76)90025-0
2. Porter, J. W. (1975), "On a theorem of Heilbronn", Mathematika , 22 (1): 20–28, doi:10.1112/S0025579300004459, MR   0498452 .