434 
History of the Theory of Numbers. 
[Chap. XVIII 
Hence the number of integers ^x which are divisible by a, but not by 
a-1, a-2,. . ., 2, is 
[x/a] a-1 
(7„=s n(i-3,). 
k=ln=2 
The number ^(a:), of primes ^x and >v = [\/^] is [x]-l-2°Z20-,. Let 
Pi,. . ., Pa be the primes ^ \/a- Let p^ be the greatest prime ^ v. Then 
<,(,)+i=M_[|]_jTn{i-A[^]}, 
from which follows Legendre's formula. 
S. Minetola-^- obtained a formula to compute the number of primes 
^K = 2k + l, not presupposing a knowledge of any primes >2, by consid- 
ering the sets of positive integers n, n',. . . for which 
(2n + l)(2n' + l)^K, (2n + l)(2n' + l)(2n" + l)^i^,. . .. 
F. RogeP^^ started with Legendre's formula for the number A{z) of 
primes ^z, introduced the remainders t—\t\, and wrote i?„(z) for the sum 
of these partial remainders. He obtained relations between values of the 
^'s and ^'s for various arguments z, and treated sums of such values. For 
arbitrary x's (p. 1815), 
Pn+l-l 
p=i 
summed for the primes p between 1 and the nth prime p„. By special 
choice of the x^s, we get formulas involving Euler's ^-function (p. 1818), 
and the number or sum of the divisors of an integer. See RogeP^ of Ch. XL 
G. Andreoli^^ noted that, if x is real, and F is the gamma function, 
^(x) = sin 
2(r(a:) + l)7r 
■sinVx 
is zero if and only if a: is a prime. Hence the number of primes < n is 
1 r'' ^'{x)dx 
2TnX . ^ix) 
The sum of the A:th powers of the primes <n is given asymptotically. 
M. Petrovitch^"*^ used a real function d{x, u), like 
a cos 27ra;+6 cos 2Tru — a — h, 
which is zero for every pair of integers x, u, and not if a: or w is fractional. 
Let$(x) be the function obtained from d{x, u) by taking 
u=\\+nx)]/x. 
Thus y=^{x) cuts the a:-axis in points whose abscissas are the primes. 
"=Giornale di Mat., 47, 1909, 305-320. 
»«Sitzung8ber. Ak. Wiss. Wien (Math.), 121, 1912, Ila, 1785-1824; 122, 1913, II a, 669-700. 
"^Rendiconti Accad. Lincei, (5), 21, II, 1912, 404-7. Wigert.'''*'' 
*«Nouv. Ann. Math., (4), 13, 1913, 406-10. 
