330 History of the Theory of Numbers. [Chap, xi 
If /(x)=log X, the second function (1) becomes v{x), ha\ing the value* 
log p when x is a power of the prime p, otherwise the value 0. Besides the 
resulting formulas, others are found by taking J{x) = v{x), Jacobi's symbol 
(A/x) in the theor>' of quadratic residues, and finally the number of repre- 
sentations of X by the system of quadratic forms of discriminant A. 
L. Saint-Loup^^ represented graphically the divisors of a number. 
Write the first 300 odd numbers in a horizontal line; the 300 following 
numbers are represented by points above the first, etc. Take any prime as 
17 and mark all its multiples; we get a rectilinear distribution of these mul- 
tiples, which are at the points of intersection of two sets of parallel lines. 
J. Hacks^^ proved that the number of integers ^m which are divisible 
by an nth power >1 is 
p„(m) =S g„] -2 [^„] +S [jtTi^.] - . . . , 
where the A;'s range over the primes >1 [Bougaief^]. Then yp2{fn) = 
m—p2{'m) is the number of integers ^m not divisible by a square >1, and 
^.w+^.(f)+^.(f)+...+^.([-^.) = 
m. 
A like formula holds for \p3 = 7n — p3(m), using quotients of m by cubes. 
L. Gegenbauer"" found the mean of the sum of the reciprocals of the 
A:th powers of those divisors of a term of an unlimited arithmetical progres- 
sion which are rth powers ; also the probabiUty that a term be divisible by no 
rth power; and many such results. 
L. Gegenbauer^^ noted that the number of integers 1, . . . , n not divisible 
by a Xth power is 
(2) Qx(n)= S^[5J/x(x). 
Ch. de la Valine Poussin^® proved that, if x is divided by each positive 
number ky-\-b^x, the mean of the fractional parts of the quotients has for 
x= 00 the limit 1 — C; if x is divided by the primes ^x, the mean of the 
fractional parts of the quotients has for x = co the limit 1 — C. Here C is 
Euler's constant.^ 
L. Gegenbauer^^ proved, concerning Dirichlet's^ quotients Q of the 
remainders (found on di\'iding n by 1 , 2, . . . , n in turn) by the corresponding 
divisors, that the number of Q's between and 1/3 exceeds the number of 
Q's between 2/3 and 1 by approximately 0.179n, and similar theorems. 
♦Cf. Bougaief 1" of Ch. XIX. 
"Comptes Rendus Paris, 107. 1888, 24; ficole Norm. Sup., 7, 1890, 89. 
"Acta Math., 14, 1890-1, 329-336. 
""Sitzungsber. Ak. Wien (Math.), 100, Ila, 1891, 1018-1053. 
^Ibid., 100, 1891, Ila, 1054. Denkschr. Akad. Wien (Math.), 49 I, II, 1885; 50 I, 1885. Cf. 
Gegenbauer" of Ch. X. 
"Annale.^ de la soc. ac. Bruxellea, 22, 1898, 84-90. 
"Sitzungsberichte Ak. Wiaa. Wien (Math.), 110, 1901, Ila, 148-161. 
