430 HISTORY OF THE THEOEY OF NUMBERS. [CHAP.XVHI P. de Monde*sir218 wrote Np for the number of multiples of the prime p which are < 2N and divisible by no prime < p. Then the number_of primes <2N is N— ?ZNp+n+I, where n is the number of primes <V2N. Also, where a, b, . . . are the primes <p. By this modification of Legendre's formula, he computed the number 78490 of primes under one million. *L. Lorenz219 discussed the number of primes under a given limit. Paolo Paci220 proved that the number of integers ^n divisible by a prime <\Sn is 2-3-5...p where r, s, . . . range over all the H primes 2, 3, . . . , p less than \/n. Thus there are n— I —N+H primes from 1 to n. The approximate value of N is l Tl , 1 L »(2-3...p)] rf—2/—h-. - H^i 1— rt o-------- r rs^ J 1 2*3...p J K. E. Hoffmann221 denoted by N the number of primes <m, by X the number of distinct prime factors of ra, by JJL the number of composite integers <m and prime to ra. Evidently JV==^>(lf)— /-i-f-A. To find AT it suffices to determine M» To that end he would count the products <ra by twos, by threes, etc. (with repetitions) of the primes not dividing ra. J. P. Gram222 proved that the number of powers of primes ^n is [Cf. Bougaief.217] Of the two proofs, one is by inversion from E. Cesaro223 considered the number x of primes ^qn and >n, where q is a fixed prime. Let «i, . . . , co, be the primes ^ n other than 1 and q. Letqk^n<qk+\ Then Let Zr>a be the number of the [qn/(&i . . .o>s)] which give the remainder r when divided by q. Set t.=2jljtt. Then 118Assoc. fran§. av. sc., 6, 1877, 77-92. Nouv. Corresp. Math., 6, 1880, 256. "Tideskr. for Math., Kjobenhavn, (4), 2, 1878, 1-3. MOSul numero de numeri primi inferior! ad un dato numero, Parma, 1879, 10 pp. ^Archiv Math. Phys., 64, 1879, 333-6. ««K. Danske Vidensk. Selskabs. Skrifter, (6), 2, 1881-6, 183-288; resume* in French, 289-308. See pp. 220-8, 296-8. »*M6m. Soc. Sc. Liege, (2), 10, 1883, 287-8.