132 HlSTOKY OF THE THEORY OF NUMBEBS. [CHAP. V where Jt(d) is Jordan's200 function. S. Schatunowsky78 proved that 30 is the largest number such that all | smaller numbers relatively prime to it are primes. He employed Tcheby- I chef's261 theorem of Ch. XVIII that, if a> 1, there exists at least one prime I between a and 2a. Cf. Wolfskehl,91 Landau,92'113 Maillet,93 Bonse,106 Remak.112 E. W. Davis76 used points with integral coordinates ^0 to visualize and prove (1) and (4). K. Zsigmondy77 wrote r, for the greatest integer ^r/s and proved that, if a takes those positive integral values ^r which are divisible by no one of the given positive integers n1? . . . , np which are relatively prune hi pairs, n,*' *-l n, n', . . . ranging over the combinations of ni} . . ., np taken 1, 2, ... at a tune. Taking /(fc) = 1, we obtain for the number ^(r; %,..., np) of integers ^r, which are divisible by no one of n1? . . ., np, the expression (5) obtained by Legendre for the case in which the n's are all primes. By induction from p to p+l> we get <t>(r ; ni, . . . , np, ?i, . . . , z/p) =<^(r; nlf . . . , wp) -S</>(rr ; %,..., np) j, . . . , np) + S 0(rn<; nx, . . . , n«.i, ni+1, . . . , np) r = 2)^(re;ni,.. ., np), c where c ranges over all combinations of powers ^r of the n's. The last becomes (4) when n\, . . . , np are the different primes dividing r. These formulas for r were deduced by him in 1896 as special cases of his inversion formula (see Ch. XIX). J. E.'Steggall78 evaluated <t>(ri) by the second method of Crelle.17 P. Bachmann79 gave an exposition of the work of Dirichlet,21 Mertens,88 Halphen54 and Sylvester55 on the mean of <t>(ri), and (p. 319) a proof of (5). L. Goldschmidt80 gave an evaluation of <£(n) by successive steps which may be combined as follows. Let p be a prime not dividing k. Each of 7lSpaczinskia Bote (phys. math.), 14, 1893, No. 159, p. 65; 15, 1893, No. 180, pp. 276-8 (Russian). "Amer. Jour. Math., 15, 1893, 84. "Jour, fiir Math., Ill, 1893, 344-6. "Proc. Edinburgh Math. Soc., 12, 1893-4, 23-24. "Die Analytische Zahlentheorie, 1894, 422-430, 481-4. «°Zeitachrift Math. Phva .. 29. 1R94. 203-4.