The asymptotic distribution of the order of elements in symmetric semigroups

P. Erdös and P. Turán [8] (Acta Math. Acad. Sci. Hungar., 18 (1967), 309–320) have shown that, if K(n, x) is the number of elements P in S_n, the symmetric group on n letters, whose order O(P) satisfies

$\text{log}\text{O(P) ?}\text{1}\text{2}\text{log}{}^2\text{n + (}\text{1}\text{3}\text{) x}\text{log}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{n}$

then

In this paper the analogous result for the symmetric semigroup is obtained. Let α?T_n, the symmetric semigroup on n letters (the set of all mappings of {1, 2,…, n} into {1, 2,…, n}) and let O(α) be the order of α. If L(n, x) is the number of α?T_n with

$\text{log}\text{O(α)?}\text{1}\text{8}\text{log}{}^2\text{n +(?}\text{1}\text{24}\text{) x}\text{log}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{n,}$

then