126 History of the Theory of Numbers. [Chap. V 
Franz Walla^- considered the product P of the first n primes > 1. Let 
a*i, . . . , X, be the integers <P/2 and prime to P, so that v=4>{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 — 2Xj (j = 1, . . . , v) are the a:'s in some order. 
Half of the a:'s are <P/4. 
J. Perott^^ proved that 
the context showing that the summations extend over all the primes p< for 
which Kpi^N [Lucas"]. He proved that 
lim ^jN) _ 3 
iV = «) m 7r2 
and gave a table showing the approximation of SN^/tt^ to $(iV) for iV^ 100. 
The last formula, proved earlier by Dirichlet^^ and Mertens,^^ was proved 
by G. H. Halphen^^ by the use of integrals and f -functions. 
Sylvester^*" defined the frequency 5 of a divisor d of one or more given 
integers a, h, . . . , I to he the number of the latter which are divisible by 
d. By use of (4) he proved the generalization 
X8(f>{d)=a+h-\-...-\-l. 
d 
J. J. Sylvester^^ stated that the number of [irreducible proper] fractions 
whose numerator and denominator are ^j is T{j) = <f){l)+ . . . +<t>{j), and 
that 3 PoT •'" f-'/^] n^-\-i 
stU^s S(/,(t)=^, 
k=i L/CJ ^=1 i=i ^ 
whence T{j)/f approximates S/tt^ as j increases indefinitely. 
If u{x) denotes the sum of all the integers <x and prime to x, and if 
U(j)=u(l)+ . . .-{-u(j), then U{j) is the sum of the numerators in the 
above set of fractions, and* 
When j increases indefinitely, U{j)/f approximates I/tt^. For each integer 
n^ 1000 the values of (^(n), T{n), Srr/ir'^ are tabulated, 
Sylvester^^ stated the preceding results and noted that the first formula 
is equivalent to 
!I3^^^) 
l(/+i). 
"Archiv Math. Phys., 66, 1881, 353-7. 
"Bull, des Sc. Math, et Astr., (2), 5, I, 1881, 37-40. 
"Comptes Rendua Paris, 96, 1883, 634-7. 
"«Amer. 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." 
"Comptes Rendus Paris, 96, 1883, 409-13, 463-5; Coll. Math. Papers, 4, 84-90. Proofs by 
F. Rogel and H. W. Curjel, Math. Quest. Educ. Times, 66, 1897, 62-4; 70, 1899, 56. 
*With denominator 3, but corrected to 6 by Sylvester," which accords with Ces&,ro." The 
editor of Sylvester's Papers stated in both places that the second member should be 
jij + l){2j+l)/12, evidently wrong for; =2. 
