308 HISTORY OF THE THEOEY OF NUMBERS. C. A. Laisant101 considered the number nk(N) of ways N can be expressed as a product of k factors (including factors unity), counting PQ . . . and QP . . . as distinct decompositions. Then nk(N) ^ E. Cesaro102 proved Gauss' result that the number of divisors, not squares, of n is asymptotic to 67r~"2 log ft. Hence r(n2) is asymptotic to Sx""2 log2^. The number of decompositions of n into two factors whose g. c. d. has a certain property is asymptotic to the product of log n by the probability that the g. c. d. of two numbers taken at random has the same property. E. Busche103 gave a geometric proof of his95 formula. But if we take $(x) to be a continuous function decreasing as x increases, with $(0)>0, then the number of positive divisors of y which are ^^(y) is 2[4>(aO/a;], summed f or x = 1, 2, . . . , with <£(x) ^ 0. This result is extended to give the number of non-associated divisors of y+zi whose absolute value is ^<#>(t/, z). J. W. L. Glaisher104 considered the excess H(n) of the number of diivisors = 1 (mod 3) of n over the number of divisors =2 (mod 3), proved that H (pq) =H(p)H(q) it p, q are relatively prime, and discussed the relation of H(ri) to Jacobi's11 E(ri). Glaisher106 gave recursion formulae for H(ri) and a table of its values for n = l,...,100. L. Gegenbauer106 found the mean value of the number of divisors of an integer which are relatively prime to given primes PI , . . . , pf, and are also (pr)th powers and have a complementary divisor which is divisible by no rth powers. Also the mean of the sum of the reciprocals of the kth powers of those divisors of an integer which are prime to pl9 . . . , pff and are rth powers. Also many similar theorems. Gegenbauer106a expressed SJI8 F(4x+l) and SF(4z-f 3) in terms of Jacobi's symbols (A/7/) and greatest integers [y] when F(x) is the sum of the kth powers of those divisors ^ \/x of x which are prime to D, or are divisible by no rth power > 1, etc.; and gave asymptotic evaluations of these sums. J. P. Gram107 considered the number Dn(m) of divisors ^m of n, the number iV2,3...W of integers ^n which are products of powers of the primes 2, 3, . . , and the sum L2, 3. . . (n) of the values of \(k) whose arguments k are the preceding N numbers, where X(2a3/9 . . . ) = ( — l)a+/9+- • • If p = Pilfp£* . . . , where the pi are distinct primes, JD,(n) =N(n) - 1MBuU. Soc. Math. France, 16, 1888, 150. 102Atti R. Accad. Lincei, Rendiconti, 4, 1888, I, 452-7. "3 Jour, fur Math., 104, 1889, 32-37. 1MProc. London Math. Soc., 21, 1889-90, 198-201, 209. 10*Ibid., 395-402. See Glaisher."1 i<*Denkschriften Ak. Wiss. Wien (Math.), 57, 1890, 497-530. 106QSitzungsber. Ak. Wiss. Wien (Math.), 99, 1890, Ha. 390-9.