308 History of the Theory of Numbers. [Chap, x 
C. A. Laisant^"^ 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 
n,{N) =n,_,{N)u(l-\-^y N=Upr. 
E. Ces^ro^"^ proved Gauss' result that the number of di\'isors, not 
squares, of n is asymptotic to Gtt"^ logn. Hence T{n~) is asymptotic to 
Stt"^ log^n. 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. Busche^''^ gave a geometric proof of his^^ formula. But if we take 
$(x) to be a continuous function decreasing as x increases, with $(0)>0, 
then the number of positive divisors of y which are ^4^iy) is S[$(a:)/x], 
summed for x = 1, 2, . . . , with $(x) ^ 0. This result is extended to give the 
number of non-associated di\'isors of y+zi whose absolute value is ^4>{y, z). 
J. W. L. Glaisher^^ considered the excess H{n) of the number of divisors 
= 1 (mod 3) of n over the number of divisors =2 (mod 3), proved that 
H{pq) =H{p)H{q) if p, q are relatively prime, and discussed the relation of 
H{n) to Jacobi's^i E{n). 
Glaisher^^^ gave recursion formulae for H{n) and a table of its values for 
n = l,..., 100. 
L. Gegenbauer^°^ found the mean value of the number of divisors of an 
integer which are relatively prime to given primes pi, . . . , Pa, and are 
also (pr) th powers and have a complementary divisor which is di\'isible 
by no rth powers. Also the mean of the sum of the reciprocals of the A;th 
powers of those di\'isors of an integer which are prime to pi, . . . , p„ and are 
rth powers. Also many similar theorems. 
Gegenbauer^o^" expressed S^=S i^(4.T+l) and SF(4a;+3) in terms of 
Jacobi's symbols (A/?/) and greatest integers [ij] when F{x) is the sum of the 
A;th powers of those divisors ^ -x/x of x which are prime to D, or are divisible 
by no rih. power > 1, etc.; and gave asymptotic evaluations of these sums. 
J. P. Gram^°^ considered the number D„(m) of di\dsors ^m of n, the 
number iV"2, 3 ..(^) of integers ^n which are products of powers of the 
primes 2, 3, . . , and the sum -Lo. 3. . . (n) of the values of \(k) whose arguments 
k are the preceding N numbers, where X(2"3^ . . . ) = ( ~ 1)"+'^+ •. 
If p = Pi°'P2°'. . ., where the p, are distinct primes, 
Dj,{n) = Nin) -'LN(n/p,''^+') +SiV(n/pi"'+ ^2'^'^') - • • • • 
""Bull. Soc. Math. France, 16, 1888, 150. 
i^Atti R. Accad. Lincei, Rendiconti, 4, 1888, I, 452-7. 
>MJour. fur Math., 104, 1889, 32-37. 
"xProc. London Math. Soc, 21, 1889-90, 198-201, 209. ^°Hbid., 395-402. See Glaisher.'" 
'»«Denkschi-iften Ak. Wiss. Wien (Math.), 57, 1890, 497-530. 
'""'Sitzungsber. Ak. Wiss. Wien (Math.), 99, 1890, Ila, 390-9. 
'"Det K. Danske Videnskab. Selskaba Skrifter (natur. og math.), (6), 7, 1890, 1-28, with r6- 
8um6 in French, 29-34. 
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