CHAP, xviii] NUMBER OF PRIMES. 433 since the only real zeros of f(x) are the primes. The integration extends over a closed contour enclosing the segment of the x-axis from 1 to n and narrow enough to contain no complex zero of f(x) . T. Levi-Civita2366 gave an analytic formula, involving definite integrals and infinite series, for the number of primes between a and ft. L. Gegenbauer237 gave formulas, similar to that by von Koch,235 for the number of primes 4s=*=l or 6s=*=l which are ^n. A. P. Minin2370 wrote \l/(y) = 0 or 1 according as y is composite or prime; then 6(n-l) as[»-2]+[n-5]+[n-7]+ . . . -^(*-l)[n-a?], summed for all composite integers x. Gegenbauer2375 proved that Sylvester's231 expression for the number of primes >n and <2n equals 2J/i(o;)[m/a;+l/2], where x takes those integral values ^ 2n which are products of prunes ^ \/2n. F. Rogel238 gave a recursion formula for the number of primes gra. T. Hayashi239 wrote Rf/q for the remainder obtained on dividing/ by q. By Laurent's187 result, — RFn(x)/(xn— l)?i,n~2 = 0 or 1 according as n is composite or prime. Hence the sum of the jth powers of the primes between s and t is t « / _\ which becomes the number of primes for j = 0. If a is a primitive nth root of unity, Wilson's theorem shows that y-o according as n is prime or composite. Hence Rx(n~l}] / (xn — 1) = 1 or 0 according as n is prime or composite. Thus is the number of primes between s and t. Hayashi240 reproduced the second of his two preceding results and gave it the form r Jo T -* or 0, according as n is prime or not, and gave a direct proof. J. V. Pexider241 investigated the number \f/(x) of primes :g x. Write 238&Atti R. Accad. Lincei, Rendiconti, (5), 4, 1895,1, 303-9. M7Monatshefte Math. Phys., 7, 1896, 73. 237aBull. Math. Soc. Moscow, 9/1898, No. 2; Fortschritte, 1898, 165. 237faMonatshefte Math. Phys., 10, 1899, 370-3. 238Archiv Math. Phys., (2), 17, 1900, 225-237. 239Jour. of the Phya. School in Tokio, 9, 1900; reprinted in Abhand. Gesch. Math. Wiss., 28,