While formerly the research of asymptotic expressions in the theory of numbers was largely confined to the approximate determination of the summatory function (or the mean value) of a given arithmetic function, recent progress in the theory of prime numbers has opened a new field for asymptotic investigations, viz., the research of upper and lower limits of an arithmetic function for large values of its argument. The first result in this line was obtained by Landau :| ,. . e <p(x) _c .. <p(x)