CHAP. X] SUM AND NUMBER OF DIVISORS. 317 where t=3, 5, 7, . . . in 2)', while the a's range over the solutions of 01+ . . . +oB_8 = i, ai+ 2a2+ . . . +(r-3)ar_3 =r. The case n = 0 leads to relations for r(r). J. de Vries1330 proved the first formula of Lerch's.117 A. Berger134 considered the excess \f/(k) of the sum of the odd divisors of k over the sum of the even divisors and proved that ^(n)+^(n-l)+^(n-3)+^(w-6)-H'(n-10) + . . - =0 or n, according as n is not or is a triangular number; also Euler's (2). J. Franel135 employed two arbitrary functions /, g and set F(n) = 2/0), G(n) = 2 y-i .y-i where d ranges over the divisors of n. Then (?" + 2 0(r) y-i where v=[Vn]. The case /(o;)=0(z) = l gives MeissePs22 (11). Next, he evaluated S#(j), where ^(n) =2f(x)g(y)h(z), summed for the sets of positive integral solutions of xyz=n. In particular, $(n) is the number of such sets if/=0 = /& = l. Using Dirichlet's series, it is shown (p. 386) that y-i where e is of the order of magnitude of n2/3 log n, C is Euler^s constant and Ci =0.0728 . . . [Piltz,62 Landau137]. Franel136 proved that where A0 is a coefficient in a certain expansion, and ep1/2 remains in absolute value inferior to a fixed number for every p. E. Landau137 gave an immediate proof of (11) and of where T3(y) is the number of decompositions of v into three factors. He obtained by elementary methods a formula yielding the final result of Franel135 on 2T*(v). R. D. von Sterneck1370 proved Jacobi's10 formula for s3. 133aK. Akad. Wetenschappen te Amsterdam, Verslagen, 5, 1897, 223. 134Nova Acta Soc. Sc. Upsaliensis, (3), 17, 1898, No. 3, p. 26. l35Math. Annalen, 51, 1899, 369-387. лл/Zrid., 52, 1899, 536-8. 137/bid., 54, 1901, 592-601.