Chap. X] SuM AND NUMBEK OF DiVISORS. 325 
G. H. Hardy^^° proved that for Dirichlet's^^ formula (7) there exists a 
constant K such that e > Kn^^'^, e < — Kn^^'^, for an infinitude of values of n 
surpassing all limit. In Piltz's^^ formula 
S Tk{n) = x{akii\og xy-'-\- . . . -^-a^k} +e„ 
n=l 
ek>Kx\ ik<—Kx\ where t={k — \)/{2k). He gave two proofs of an 
equivalent to Voronoi's^^^'' explicit expression for T{x). 
Hardy^^^ wrote A(n) for Dirichlet's e in (7) and proved that,^° for every 
positive e, /l{n) = 0{n'^^''^) on the average, i. e., 
iJiA(0M«=O(n'+"*). 
G. H. Hardy and S. Ramanujan^^^ employed the phrase "almost all 
numbers have a specified property" to mean that the number of the num- 
bers ^ X having this property is asymptotic to a: as a; 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-\-h different prime factors, 
where a = log log n, h=f-\/d. 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{n) is defined to mean f{n) 
if , for every positive e, (1— €)/(n)<gr(n)<(l+e)/(n) for almost all values 
of n. 
S. Wigert^^^ gave an asymptotic representation for l!,n^j:r{n){x — n)^. 
E. T. BelP^ gave results bearing on this chapter. 
F. RogeP^^ expressed the sum of the rth powers of the divisors ^g* of 
m as an infinite series involving Bernoullian functions. 
A. Cunningham^^^ found the primes p< lO'* (or 10^) for which the number 
of divisors of p — 1 is a maximum 64 (or 120). 
Hammond^^ of Ch. XI and RogeP^^ of Ch. XVIII gave formulas involv- 
ing (J and r. Bougaief^^' ^^ of Ch. XIX treated the number of divisors 
^ m of n. Gegenbauer^° of Ch. XIX treated the sum of the pth powers of 
the divisors ^ m of n. 
i^oProc. London Math. Soc, (2), 15, 1916, 1-25. 
i"/&id., 192-213. 
i82Quar. Jour. Math., 48, 1917, 76-92. 
i83Acta Math., 41, 1917, 197-218. 
is^Annals of Math., 19, 1918, 210-6. 
i85Math. Quest. Educ. Times, 72, 1900, 125-6. 
i86Math. Quest, and Solutions, 3, 1917, 65. 
