126 HlSTOEY OF THE THEORY OP NUMBERS. [CHAP. V Franz Walla52 considered the product P of the first n primes > 1. Let «!,..., S, be the integers <P/2 and prime to P, so that v=<j>(P)/2. Then, if n>2, half of the x's are =1 (mod 4) and the others are =3 (mod 4). Also, the absolute values of £P— 2av (j = l, . . ., v) are the x's in some order. Half of the x'$ are <P/4. J. Perott53 proved that the context showing that the summations extend over all the primes p»- for which Kp^N [Lucas72]. He proved that lim &(N) 3 tf = oo N2 7T2 and gave a table showing the approximation of 3JV2/7r2 to $(N) for JVS 100. The last formula, proved earlier by Dirichlet21 and Mertens,36 was proved by G. H. Halphen64 by the use of integrals and f -functions. Sylvester540 defined the frequency d of a divisor d of one or more given integers a, b, . . . , I to be the number of the latter which are divisible by d. By use of (4) he proved the generalization ^tated that the number of [irreducible proper] fractions A denominator are ^j is T(j) — <l>(l')+. . . +#(j), and j)/j2 approximates 3/?r2 as j increases indefinitely. .) denotes the sum of all the integers <x and prime to x, and if /(!)+.. .+u(j}, then U(j) is the sum of the numerators in the act of fractions, and* When; increases indefinitely, U(j)/f approximates l/7r2. For each integer n5g 1000 the values of £(n), T(n), 3n2/7r2 are tabulated. Sylvester66 stated the preceding results and noted that the first formula is equivalent to "Archiv Math. Phys., 66, 1881, 353-7. "Bull, des Sc. Math, et Astr., (2), 5, I, 1881, 37-40. "Comptes Rendufl Paris, 96, 1883, 634-7. WaAmer. Jour. Math., 5, 1882, 124; Coll. Math. Papers, 3, 611. "Phil. Mag., 15, 1883, 251-7; 16, 1883, 230-3; Coll. Math. Papers, 4, 101-9. Cf. Sylvester.84 "Comptes Rendus Paris, 96, 1883, 409-13, 463-5; Coll. Math. Papers, 4, 84-90. Proofs by F. Rogeland H. W. Curjel, Math. Quest. Educ. Times, 66, 1897, 62-4; 70, 1899, 56. *With denominator 3, but corrected to 6 by Sylvester,60 which accords with Cesaro.85 The editor of Sylvester's Papers stated in both places that the second member should be j'C?-t-l)(2j+l)/12, evidently wrong/or j=2.