132 History of the Theory of Numbers. [Chap, v 
2Q©d'=2pg)/,(d), 
where Jt{d) is Jordan's-"" function. 
S. Schatunowsky"* proved that 30 is the largest number such that all 
smaller numbers relatively prime to it are primes. He employed Tcheby- 
chef's^" theorem of Ch. XVIII that, if a> 1, there exists at least one prime 
between a and 2a. Cf. Wolfskehl,^^ Landau,^^. 113 Maillet,^^ Bonse/"« 
Remak.^^2 
E. W. Da\ds''^ used points with integral coordinates ^0 to visualize 
and prove (1) and (4). 
K. Zsigmondy^^ wrote r, for the greatest integer ^ r/s and proved that, 
if a takes those positive integral values ^r which are di\asible by no one 
of the given positive integers rii, . . . , n^ which are relatively prime in pairs, 
r rn rnn' 
2/(a) = S f{k) -S S f{kn) +22 f{knn') -..., 
k=l n k=l n, n' k = l 
n, n',. . . ranging over the combinations of rii,. . ., n^ taken 1, 2, . . . at a 
time. Taking /(A:) = 1, we obtain for the number (f>{r; rii, . . . , nj of integers 
^r, which are divisible by no one of ni, . . . , n^, the expression (5) obtained 
by Legendre for the case in which the n's are all primes. By induction 
from p to p+1, we get 
+ 2<^(r^;ni,. ..,n,)-..., 
p 
r=4>{r] ni, . . . , n,)+ 2 </)(r„<; rii, . . . , n,_i, n,+i, . . . , nj 
t=i 
+ 2<^(r„„/; riiST^n, n') + 
r = 20(r,;ni,.. ., n,), 
where c ranges over all combinations of powers ^r of the n's. The last 
becomes (4) when ni,. . ., n^ are the different primes di\ading r. These 
formulas for r were deduced by him in 1896 as special cases of his inversion 
formula (see Ch. XIX). 
J. E. Steggair^ evaluated </)(n) by the second method of Crelle.^^ 
P. Bachmann^^ gave an exposition of the work of Dirichlet,^^ Mertens," 
Halphen^ and Sylvester^^ on the mean of <p{n), and (p. 319) a proof of (5). 
L. Goldschmidt^" gave an evaluation of <j){n) by successive steps which 
may be combined as follows. Let p be a prime not dividing k. Each of 
"Spaczinakis Bote (phys. math.), 14, 1893, No. 159, p. 65; 15, 1893, No. 180, pp. 276-8 
(Russian). 
"Amer. Jour. Math., 15, 1893, 84. 
"Jour, fur Math., Ill, 1893, 344-6. 
"Proc. Edinburgh Math. Soc, 12, 1893-4, 23-24. 
"Die Anab-tische Zahlentheorie, 1894, 422^30, 481-4. 
««Zeitschrift Math. Phys., 39, 1894, 203-4. 
