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CHAP. XVHI]            ASYMPTOTIC DISTRIBUTION OF PRIMES.                          439
A. de Polignac305 crossed out the multiples of 2 and 3 from the series of natural numbers and obtained the " table a®":
(0)  1   (2)   (3)   (4)   5  (6)   7   (8)    (9)    (10)    11....
The numbers of terms in the successive sets of consecutive deleted numbers are 1, 3, 1, 3, 1, . . . , which form the "diatomic series of 3." Similarly, after deleting the multiples of the first n primes, we get a table an and the diatomic series of the nth prime Pn. That series is periodic and the terms after 1 of the period are symmetrically distributed (two terms equidistant from the ends are equal), while the middle term is 3. Let irn denote the product of the primes 2, 3, . . . , Pn. Then the number of terms in the period is $(7rn). The sum of the terms in the period is irn— <l>(irn) and hence is the number of integers <irn which are divisible by one or more primes ^Pn. As applications he stated that there exists a prime between Pn and Pn2, also between an and an+1. He306 stated that the middle terms other than 3 of a diatomic series tend as n increases to become 1, 3, 7, 15, . . . , 2m— 1, . . . .
J. Deschamps307 noted that, after suppressing from the series of natural numbers the multiples of the successive primes 2, 3, . . . , p, the numbers left form a periodic series of period 2-3... p] and similar theorems. Like remarks had been made previously by H. J. S. Smith.308
P. L. Tchebychef s261 investigation shows that for x sufficiently large the number TT(X) of primes ^x is between 0-921Q and M06Q, where Q = z/log x. He314 proved that the limit, if existent, of ir(x)/Q for cc = oo is unity. J. J. Sylvester267 obtained by the same methods the limits 0-95Q and 1-05Q.
By use of the function f (s) =2""f n"8 of Biemann, J. Hadamard31* and Ch. de la Valle'e-Poussin316 independently proved that the sum of the natural logarithms of all primes ^x equals x asymptotically. Hence follows the fundamental theorem that ir(x) is asymptotic to Q, i. e.,
806Recherchea nouvelles sur lea nombres premiers, Paris, 1851, 28 pp. Abstract in Comptes Rendus Paris, 29, 1849, 397-401, 738-9; same in Nouv. Ann. Math., 8, 1849, 423-9. Jour, de Math., 19, 1854, 305-333.
»MNouv. Ann. Math., 10, 1851, 308-12.
•"Bull. Soc. Philomathique de Paris, (9), 9, 1907, 102-112
•MProc. Aflhmolean Soc., 3,1857, 128-131; Coll. Math. Papers, 1, 36.
'"Mem. Ac. Sc. St. P6tersbourg, 6, 1851, 146; Jour, de Math., 17, 1852, 348; Oeuvres, 1, 34.
""Bull. Soc. Math, de France, 24, 1896, 199-220.
""Annales de la Soc. Sc. de Bruxelles, 20, II, 1896, 183-256.