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Full text of "History Of The Theory Of Numbers - I"

434                         HlSTOBY OF THE THEORY OF NuMBEKS.            [CHAP. XVIII
Hence the number of integers £x which are divisible by a, but not by a — 1, a— 2,. .., 2, is
[*/a] o-l
aa=s n (!-«„).
>(»), of primes ^x and >? = [Vz] is M — 1— SIIJ0-..   Let 'ie primes ^ Va«   Let P« be the greatest prime g v.   Then
fan   " t'&'^L   ,rwfc~n -i-M- « -s 2 nn-A ^- [.
L^J     o»3Ai=l /i=2L          LPMJJ
jegendre's formula.
otained a formula to compute the number of primes >resupposing a knowledge of any primes >2, by consid->ositive integers n, n', . . . for which
yd with Legendre's formula for the number A (z) of d the remainders t—\t\, and wrote Rn(z) for the sum inders. He obtained relations between values of the is arguments z, and treated sums of such values. For •OAO),
Pn-fl-1
S'   (xv+1 - xv)A (v) = -2xp+xPn+1 A (pn) ,
l"»l
summed for the primes p between 1 and the nth prime pn.   By special
choice of the z's, we get formulas involving Euler's ^-function (p. 1818),
and the number or sum of the divisors of an integer.    See Rogel22 of Ch. XI.
G. Andreoli244 noted that, if x is real, and T is the gamma function,
X
is zero if and only if x is a prime.   Hence the number of primes <n is
The sum of the fcth powers of the primes <n is given asymptotically. M. Petrovitch245 used a real function 6(x, u), like
a cos 2irx-\-b cos 2iru—a — b,
which is zero for every pair of integers x, u, and not if x or u is fractional. Let$(x) be the function obtained from 6(x, u) by taking
u={l+T(x)}/x. Thus 2/=$(x) cuts the re-axis in points whose abscissas are the primes.
'"Giornale di Mat., 47, 1909, 305-320.
M'Sitzungsber. Ak. Wiss. Wien (Math.), 121, 1912, II a, 1785-1824; 122, 1913, II a, 669-700.
2«Rendiconti Accad. Lincei, (5), 21, II, 1912, 404-7.    Wigert.236a
"BNouv. Ann. Math., (4), 13, 1913, 406-10.