Porter's Constant

Porter's constant is the constant appearing in formulas for the efficiency of the Euclidean algorithm,

			(1)
			(2)
			(3)

(OEIS A086237), where  is the Euler-Mascheroni constant,  is the Riemann zeta function, and  is the Glaisher-Kinkelin constant (Knuth 1998, p. 357). The notation  is generally used for this constant (Knuth 1998, p. 357, Finch 2003, pp. 156-157), though other authors use  (Ustinov 2010) or  (Dimitrov et al. 2000).

The related constant originally considered by Porter (1975) and Knuth (1976) was denoted  and , respectively, and defined by

			(4)
			(5)

Knuth (1976) suggested  be called the Lochs-Porter constant due to the work of Lochs (1961).

REFERENCES:

Dimitrov, V. S.; Jullien, G. A.; and Miller, W. C. "Complexity and Fast Algorithms for Multiexponentiations." IEEE Trans. Comput. 49, 141-147, 2000.

Finch, S. R. "Porter-Hensley Constants." §2.18 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 156-160, 2003.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 113, 2003.

Knuth, D. E. "Evaluation of Porter's Constant." Computers Math. Appl. 2, 137-139, 1976.

Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998.

Lochs, G. "Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmässigen Kettenbrüche." Monatsh. f. Math. 65, 27-52, 1961.

Porter, J. W. "On a Theorem of Heilbronn." Mathematika 22, 20-28, 1975.

Sloane, N. J. A. Sequence A086237 in "The On-Line Encyclopedia of Integer Sequences."

Ustinov, A. V. "The Mean Number of Steps in the Euclidean Algorithm with Odd Partial Quotients." Math. Notes 88, 574-584, 2010.