318 History of the Theory of Numbers. [Chap, x 
J. Franel"^ stated that, if /(n) is the number of positive integral solutions 
of x''y^ = n, where a, h are distinct positive integers, 
2/(r) =^i^l)n^+^{^n'+o{n'-^ 
+6J, 
where'" 0{s) is of the order of magnitude of s. Taking a = l, 6 = 2, we see 
that /(n) is the number of di\isors of q, where <f is the greatest square divid- 
ing n, and that the mean of /(n) is tt"/ 6. 
E. Landau^^^ proved the preceding formula of Franel's. 
EUiott^^ of Ch, V gave formulas involving a{n) and rin). 
L. Kronecker^'*'^ proved that the sum of the odd divisors of a number equals 
the algebraic sum of all its di\isors taken positive or negative according as 
the complementary^ di\'isor is odd or even (attributed to Euler^); proved 
(pp. 267-8) the result of Dirichlet'^ and (p. 345) proved (7) and found the 
median value (Mittelwert) of T{n) to be log^ n+2C with an error of the order 
of magnitude of n~^^^when the number of values employed is of the order of 
n^''^. Calling a di\isor of n a smaller or greater di\dsor according as it is less 
than or greater than \/n, he found (pp. 343-369) the mean and median value 
of the sum of all smaller (or greater) di\'isors of 1, 2, . . . , A^ [cf. Gegenbauer^^], 
the sum of their reciprocals, and the sum of their logarithms. The mean of 
Jacobi's^^ E{n) is x/4 (p. 374). 
J. W. L. Glaisher^^^ tabulated for n = l,..., 1000 the values of the 
function^'^ H{n) and of the excess J{n) of the number of divisors of n which 
are of the form 8A- + 1 or 8A;+3 over the number of divisors of the form 
8A:+5 or 8A:+7. WTien n is odd, 2J{n) is the number of representations of n 
by x'-\-2f. 
J. W. L. Glaisher"^ derived from Dirichlet's^^ formula, and also inde- 
pendently, the simpler formula 
2 0.(s) = -pG(p)+f[5],(.)+£(?{g]}, 
where p = {\/n]. The case g{s) = l gives Meissel's^- formula (11), which is 
applied to find asymptotic formulae involving n/s — [n/s]. The error of the 
approximation (7) is discussed at length (pp. 38-75, 180-2). The first 
formula above is applied (pp. 183-229) to find exact and asymptotic formu- 
las for 2/(s), when/(n) is Jacobi's" E{n), Glaisher's^^^ H{n) or J{n), or the 
excess D{n) of the number of odd divisors of n over the number of even 
di\'isors, or more general functions (p. 215, p. 223) involving the number of 
di\isors with specified residues modulo r. 
G. Voronoi^^^ proved a formula like Dirichlet's^^ (7), but with e now of the 
same order of magnitude as -^/n log^ n. 
•"L'intermddiaire des math., 6, 1899, 53; 18, 1911, 52-3. 
"»/6ui., 20, 1913, 155. 
•"Vorlesungen iiber Zahlentheorie, I, 1901, 54-55. 
"'Messenger Math., 31, 1901-2, 64-72, 82-91. 
>«Quar. Jour. Math., 33, 1902, 1-75, 180-229. 
'«Jour. fiir Math., 126, 1903, 241-282. 
