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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
. . .+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.