Chap. XI] MISCELLANEOUS THEOREMS ON DIVISIBILITY. 329 
Gegenbauer^°" proved that the arithmetical mean of the greatest integers 
contained in k times the remainders on the division of n by 1, 2, . . ., n 
approaches 
k—l 
k\ogk-{-k-l-ki:i/x 
as n increases. The case A; = 2 is due to Dirichlet. 
Gegenbauer^^ gave formulas involving the greatest divisor t^n), not 
divisible by a, of the integer n. In particular, he gave the mean value of 
the greatest divisor not divisible by an ath power. 
L. Gegenbauer,^^ employing Merten's function ix (Ch. XIX) and 
R{a)=a — \a\, gave the three general formulas 
2 sV(^V(2/) = sV(A;) - i: m - i m, 
Xi j/=i \y / k=i A=i it=i 
where X2 ranges over the divisors >n of (r — l)n+l, (r — l)n+2, . . ., rn, 
while Xi ranges over all positive integers for which 
r-\-n ~ g r n \ ' g' ' ' gj 
where g is the g. c. d. of r, n. Take f{x) = 1 or according as x is an sth 
power or not. Then the functions 
(1) 2 /(A;), 2/x(^)/(2/) 
k = \ y = \ \y/ 
become [-^m] and \{^), with the value if the exponent of any prime 
factor of X is ^0, 1 (mod s), otherwise the value ( — 1)", where a is the 
number of primes occurring in x to the power /cs+1. Thus 
2x,(x2) = \y^ - \</V^r?^ - [i/i\ • 
If j{x) = or 1 according as x is divisible by an sth power or not, the func- 
tions (1) become Qs(w) and ix{\/x)j the former being the number of integers 
^ m divisible by no sth power. If J{x) = 1 or according as x is prime or 
not, the functions (1) become the number of primes ^m and a simple func- 
tion a(x) ; then the third formula shows that the mean density of the primes 
loiDenkschr. Akad. Wien (Math.), 49, II, 1885, 108. 
"Sitzungsber. Akad. Wiss. Wien (Math.), 94, 1886, II, 714. 
i276id., 97, 1888, Ila, 420-6. 
