Chap. XVIII] ASYMPTOTIC DiSTKIBUTION OF PRIMES. 439 
Diatomic Series. 
A. de Polignac^°^ crossed out the multiples of 2 and 3 from the series of 
natural numbers and obtained the "table 02": 
(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 a„ and the dia- 
tomic series of the nth prime P„. 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 7r„ denote the product 
of the primes 2, 3, . . . , P^- Then the number of terms in the period is 
0(7r„). The sum of the terms in the period is x„— 0(7r„) and hence is the 
number of integers <7r„ which are divisible by one or more primes ^Pn- 
As applications he stated that there exists a prime between P„ and P^, also 
between o" and a""^\ He^°^ stated that the middle terms other than 3 of a 
diatomic series tend as n increases to become 1, 3, 7, 15, . . . , 2"* — 1, . . . . 
J. Deschamps^°^ 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.^°^ 
Asymptotic Distribution of Primes. 
P. L. Tchebychef's^^^ investigation shows that for x sufficiently large 
the number 7r(a;) of primes ^a; is between 0-921Q and 1-106Q, where 
Q = a:/log X. He^^^ proved that the limit, if existent, of 7r(x)/Q for x= 00 is 
unity. J. J. Sylvester^^^ obtained by the same methods the limits 0-95Q 
and 1-05Q. 
By use of the function f (s) =2"zfn~* of Riemann, J. Hadamard^^^ 
and Ch. de la Vallee-Poussin^^^ independently proved that the sum of the 
natural logarithms of all primes ^x equals x asymptotically. Hence 
follows the fundamental theorem that -wix) is asymptotic to Q, i. e., 
. log a : 
lim t:{x) — ^— =1. 
'°*Recherches nouvelles sur les 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. 
»«Nouv. Ann. Math., 10, 1851, 308-12. 
8"Bull. Soc. Philomathique de Paris, (9), 9, 1907, 102-112 
3»«Proc. Ashmolean Soc, 3, 1857, 128-131; Coll. Math. Papers, 1, 36. 
3"M6m. Ac. Sc. St. P^tersbourg, 6, 1851, 146; Jour, de Math., 17, 1852, 348; Oeuvres, 1, 34. 
"6BuU. Soc. Math, de France, 24, 1896, 199-220. 
"«Annales de la Soc. Sc. de Bruxelles, 20, II, 1896, 183-256. 
