Постоянная Голомба — Дикмана

Постоянная Голомба — Дикмана — математическая константа, возникающая в случайных перестановках и в теории чисел, равная^[1]:

\lambda =0{,}62432998854355087099293638310083724\dots

Названа по именам Соломона Голомба и Карла Дикмана. Вычисляется из всех перестановок множества из $n$ элементов с использованием средней длины наиболее длинного цикла перестановки $a_{n}$ :

\lambda =\lim _{n\to \infty }{\frac {a_{n}}{n}}

С точки зрения теории вероятностей $\lambda n$ является асимптотой ожидания длины наиболее длинного цикла равномерно распределённых случайных перестановок множества из $n$ элементов.

В теории чисел постоянная возникает в связи со средним значением наибольшего простого делителя целого числа:

\lambda =\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=2}^{n}{\frac {\log(P_{1}(k))}{\log(k)}}

где $P_{1}(k)$ — наибольший простой делитель числа $k$ . Таким образом, если $k$ — $d$ -значное десятичное целое, то $\lambda d$ является асимптотой среднего числа знаков в наибольшем простом делителе $k$ .

Другой источник из теории чисел — вероятность того, что второй по величине простой делитель числа $n$ меньше квадратного корня из наибольшего простого делителя $n$ , асимптотически равная $\lambda$ :

\lambda =\lim _{n\to \infty }\operatorname {prob} \left\{P_{2}(n)\leqslant {\sqrt {P_{1}(n)}}\right\}

где $P_{2}(n)$ — второй по величине простой делитель $n$ .

Существует несколько интегральных представлений для $\lambda$ :

\lambda =\int _{0}^{\infty }e^{-t-\operatorname {Ei} _{1}(t)}dt

, где

\operatorname {Ei} _{1}(t)

— модифицированная интегральная показательная функция,

\lambda =\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}dt

\lambda =\int _{0}^{\infty }{\frac {\rho (t)}{(t+1)^{2}}}dt

, где

\rho (t)

— это функция Дикмана.

Вопрос о рациональности или иррациональности постоянной открыт.

Примечания

↑ последовательность A084945 в OEIS

Ссылки

Weisstein, Eric W. Golomb-Dickman Constant (англ.) на сайте Wolfram MathWorld.
Finch, Steven R. Mathematical Constants (неопр.). — Cambridge University Press, 2003. — С. 284—286. — ISBN 0-521-81805-2.

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