432 History of the Theory of Numbers. [Chap, xviii 
is 1/2, but the nearest integer to x in the contrary case. L. Gegenbauer^^^* 
gave a proof and generalization. 
Sylvester-^ ^'' noted that, if din) is the number of primes ^u, and if 
pi , . . . , Pi be the primes ^ \/x, and gi, . ,gy those between \/x and x, then 
xd(x/p)-m^'/q)= {e{Vx)\\ 
H. W. Curjel'-^^" noted that the number of primes >p and <p- is ^p 
if p is a prime ^5. We have only to delete from 1, 2, . . ., p^ multiples of 
2, 3, 5, . . . , or p. 
L. Gegenbauer-^- considered the integers x divisible by no square and 
formed of the odd primes ^m, when n^w^\/2n. Of the numbers [2«/x] 
which are of one of the forms 4s +1 and 4s +2, count those in which x is 
formed of an even number of primes and those in which x is formed of an 
odd number; denote the difference of the counts by a. He stated that the 
interv^al from m + 1 to n (limits included) contains a — 1 more primes than the 
inter\'al from tz+I to 2n. 
He gave (pp. 89-93) an expression for the sum of the values taken by an 
arbitrary- function g(x) when x ranges over the primes among the first n 
terms of an arithmetical i)rogression; in particular, he enumerated the 
primes ^n of the form 4s +1 or 4s — 1. 
F. Graefe-^^ would find the number of primes <m = 10000 by use of 
tables showing for each prime p, 5'^p^\/m, the values of n for which 
6n+l or 6nH-5 is divisible by p. 
P. Bachmann-^ quoted de Jonquieres,^°^ Lipschitz,-°^ Sylvester,2°^ and 
Ces^ro."^ 
H. von Koch235 wrote f{x) = (a:-l)(a:-2) . . . (x-n), 
fix) 
X=2L A J t..^=2{x-IJLv)f'{t^v) 
0(a:)=njl-^j, p{x)= 2 ,, •;.:;;,..,., (fxvSn), 
and proved that, for positive integers x^n, d(x) = 1 or according as x is 
prime or composite. The number of primes ^m^n is ^(1)+. . .+^(m). 
A. Baranowski-^^ noted the formula, simpler than Meissel's,^^^ 
xP{n)=<l>[n, ^(v^)]H-^(v^) -1 
for computing the number \f/(n) of primes ^n. 
S. Wigert^^^^ noted that the number of primes < n is 
1 C f'{x)dx , ., . . 2 , . o / i+r(x) \ 
TT^ I ./ X , where /(x) =smVx+sm-7r I ), 
2TnJ f{x) \ X / 
"'"Denkschr. Akad. Wiss. Wien (Math.), 60, 1893, 47. 
"i^Math. Quest. Educ. Times, 56, 1892, 67-8. 
"'olbid., 58, 1893, 127. 
"^Monatshefte Math. Phys., 4, 1893, 98. 
'"Zeitschrift Math. Phys., 39, 1894, 38-50. 
»»<Die Analytische Zahlentheorie, 1894, 322-5. 
"*Comptes Rendus Paris, 118, 1894, 850-3. 
"•Bull. Int. Ac. Sc. Cracovie, 1894, 280-1 (German). Cf. ♦Rozpraw'y Akad. Umiej., Cracovie, 
(2), 8, 1895, 192-219. 
'"''Ofversigt K. Vetensk. Ak. Forhand., Stockhohn, 52, 1895, 341-7. 
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