432 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xvm is 1/2, but the nearest integer to x in the contrary case. L. Gegenbauer231a gave a proof and generalization. Sylvester2316 noted that, if 6(u) is the number of primes ^w, and if pi , . . . , pi be the primes ^ \/x, and qi, . ,gy those between -\/x and x, then H. W. Curjel231c noted that the number of prunes >p and <p* is ^p if p is a prime ^5. We have only to delete from 1, 2, . . ., p2 multiples of 2,3,5,..., or p. L. Gegenbauer232 considered the integers x divisible by no square and formed of the odd primes ^w, when n^m^\/2n. Of the numbers [2n/x] which are of one of the forms 4s +1 and 4s +2, count those in which x is formed of an even number of primes and those in which x is formed of an odd number; denote the difference of the counts by a. He stated that the interval from m+ 1 to n (limits included) contains a — 1 more primes than the interval from n+l to 2n. He gave (pp. 89-93) an expression for the sum of the values taken by an arbitrary function g(x) when x ranges over the primes among the first n terms of an arithmetical progression; in particular, he enumerated the primes ^n of the form 4s+l or 4s— 1. F. Graefe233 would find the number of primes <m = 10000 by use of tables showing for each prime p, S^p^Vm, the values of n for which Qn+l or 6n+5 is divisible by p. P. Bachmann234 quoted de Jonquieres,206 Lipschitz,207 Sylvester,208 and Cesaro.223 H. von Koch235 wrote /(») = (x -!)(«- 2). . .(s-n), and proved that, for positive integers x^n, 6(x) = 1 or 0 according as x is prime or composite. The number of primes ^m^n is 0(1) + . . . +0(m). A. Baranowski236 noted the formula, simpler than Meissel's,215 for computing the number \l/(n) of primes ^n. S. Wigert236a noted that the number of primes <n is i Cf(x}dx , f, , . 2 , . 2 / TT-T I \.\ , where /(x) =sm27ra;+sm27r I 2iriJ f(x) \ M1«Denkschr. Akad. Wiss. Wien (Math.), 60, 1893, 47. M1^Math. Quest. Educ. Times, 56, 1892, 67-8. »fc/Wd., 58, 1893, 127. ^Monatshefte Math. Phys., 4, 1893, 98. M3Zeitschrift Math. Phys., 39, 1894, 38-50. ^Die Analytiache Zahlentheorie, 1894, 322-5. 236Comptes Rendus Paris, 118, 1894, 850-3. M«BuU. Int. Ac. Sc. Cracovie, 1894, 280-1 (German). Cf. *Rozprawy Akad. TJmiej., Cracovie, (2), 8, 1895, 192-219. awaOfversigt K. Vetensk. Ak. Fdrhand., Stockholm, 52, 1895, 341-7.