CHAP.X] SUM AND NlJMBEB OP DlVISOKS. 32 G. H. Hardy180 proved that for Dirichlet's17 formula (7) there exists constant K such that >Kn1/4t, < Knl/4t, for an infinitude of values of n surpassing all limit. In Piltz's52 formula n-1 . . .+akk}+ek, \ k<Kx\ where t~(k I)/(2k). He gave two proofs of an equivalent to Voronofsufia explicit expression for T(x). Hardy181 wrote A(n) for Dirichlet's 6 in (7) and proved that,90 for every positive e, A(n) = 0(nc+1/4) on the average, i. e., G. H. Hardy and S. Ramanujan182 employed the phrase "almost all numbers have a specified property" to mean that the number of the numbers ^x having this property is asymptotic to x as x increases indefinitely, and proved that if / is a function of n which tends steadily to infinity with n, then almost all numbers have between a 6 and a+b different prime factors, where a = log log n, b=f*\/a. The same result holds also for the total number of prime factors, not necessarily distinct. Also a is the normal order of the number of distinct prime factors of n or of the total number of its prime factors, where the normal order of g(ri) is defined to mean /(n) if , for every positive e, (l~-e)f(n)<g(ri)<(l+e)f(ri) for almost all values of n. S. Wigert183 gave an asymptotic representation for ^n^xr(n)(x ri)k. E. T. Bell184 gave results bearing on this chapter. F. Rogel185 expressed the sum of the rth powers of the divisors ^q of m as an infinite series involving Bernoullian functions. A. Cunningham186 found the primes p< 104 (or 105) for which the number of divisors of p I is a maximum 64 (or 120). Hammond43 of Ch. XI and Rogel243 of Ch. XVIII gave formulas involving cr and r. Bougaief69' 62 of Ch. XIX treated the number of divisors ^moin. Gegenbauer60 of Ch. XIX treated the sum of the pth powers of the divisors ^ra of n. 180Proc. London Math. Soc., (2), 15, 1916, 1-25. 181/bid., 192-213. 182Quar. Jour. Math., 48, 1917, 70-92. "'Acta Math., 41, 1917, 197-218. ^Annals of Math., 19, 1918, 210-6. 186Math. Quest. Educ. Times, 72, 1900, 125-6. "8Math. Quest, and Solutions, 3, 1917, 65.