318 HISTORY OF THE THEORY OF NUMBERS. [CHAP, x J. Franel138 stated that, if f(n) is the number of positive integral solutions of xayb=n, where a, b are distinct positive integers, where90 0(s) is of the order of magnitude of s. Taking a = 1, 6 = 2, we see that f(ri) is the number of divisors of q, where q2 is the greatest square dividing n, and that the mean of f(ri) is 7r2/6. E. Landau139 proved the preceding formula of Franel's. Elliott96 of Ch. V gave formulas involving a* (ft) and r(n). L. Kronecker140 proved that the sum of the odd divisors of a number equals the algebraic sum of all its divisors taken positive or negative according as the complementary divisor is odd or even (attributed to Euler2); proved (pp. 267-8) the result of Dirichlet15 and (p. 345) proved (7) and found the median value (Mittelwert) of r(n) to be loge n+2C with an error of the order of magnitude of n~~1/4when the number of values employed is of the order of n3/4. Calling a divisor of n a smaller or greater divisor 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) divisors of 1, 2, . . . , N [cf . Gegenbauer77], the sum of their reciprocals, and the sum of their logarithms. The mean of Jacobi's11 E(ri) is ir/4 (p. 374). J. W. L. Glaisher141 tabulated for n = l,..., 1000 the values of the function104 H(n) and of the excess J(n) of the number of divisors of n which are of the form Sk+l or 8&+3 over the number of divisors of the form 8 k +5 or 8&-f-7. When n is odd, 2J(ri) is the number of representations of n J. W. L. Glaisher142 derived from Dirichlet's17 formula, and also inde- pendently, the simpler formula S 2 0\ [£ as=i (Lo where p = [\/n]. The case g(s)=*l gives Meissel's22 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 formulas for S/(s), when/(n) is Jacobi's11 E(n), Glaisher's141 H(ri) or J(n), or the excess D(n) of the number of odd divisors of n over the number of even divisors, or more general functions (p. 215, p. 223) involving the number of divisors with specified residues modulo r. G. Vorono'i143 proved a formula like Dirichlet's17 (7), but with c now of the same order of magnitude as -\/n log, n. 138L'interm<§diaire des math., 6, 1899, 53; 18, 1911, 52-3. "9/fcid., 20, 1913, 155. 140Vorlesungen uber Zahlentheorie, I, 1901, 54-55. 141Messenger Math., 31, 1901-2, 64r-72, 82-91. l«Quar. Jour. Math., 33, 1902, 1-75, 180-229.