Chap. XVIII] NuMBER OF PRIMES. 431 
E. Catalan^^'* obtained the preceding results for the case g = 2; then ti is 
the number of odd quotients [2n/(8], (2 the number of odd quotients 
[2n/{^y)], . . . , where I3,y,. . . are the primes >2 and ^n. 
L. Gegenbauer225 gave eight formulas, (29)-(36), of the type of 
Legendre's, a special case of one being 
([^])/^(^) = 1+L,(n), SM^^^ t\ 
where x ranges over the integers divisible by no prime > Vn, while fi(x) is 
Merten's function (Ch. XIX) and Lh{n) is the sum of the kth. powers of all 
primes >\/n but Sn. The case A; = is Legendre's formula. The case 
A; = l is Sylvester's^"^ 
E. MeisseP^^ computed the number of primes < 10^ 
Gegenbauer-^*^" gave compUcated expressions for d{n), one a generaliza- 
tion of Bougaief's.^^'^ 
A. Lugli^" wrote 4>{n, i) for the number of integers ^n which are divis- 
ible by no one of the first i primes Pi = 2, p2 = 3, . . . . If 2; is the number of 
primes S -y/n and if s is the least integer such that 
the number \l/{n) of primes ^n, excluding 1, is proved to satisfy 
This method of computing \l/{n) is claimed to be simpler than that by 
Legendre or Meissel. 
J. J. van Laar^"" found the number of primes < 30030 by use of the 
primes <1760. 
C. Hossfeld^^^ gave a direct proof of 
^{gVi- ■ ■Pn=^r,n)==g{pi-l). . .(p„-l)±$(r, n), 
the case of the upper signs being due to Meissel. ^^^ 
F. Rogel"^^ gave a modification and extension of Meissel's^^^ formula. 
H. Scheffler^^'' discussed the number of primes between p and q. 
J. J. Sylvester^^^ stated that the number of primes >n and <2n is 
a ab abc 
ii a,h,. . . are the primes ^ \/2n and Hx denotes x when its fractional part 
224Mem. Soc. Sc. Lidge, M6m. No. 1. 
225Sitzungsber. Ak. Wiss. Wien (Math.), 89, II, 1884, 841-850; 95, II, 1887, 291-6. 
22«Math. Annalen, 25, 1885, 251-7. 
""ositzungsber. Ak. Wiss. Wien (Math.), 94, II, 1886, 903-10. 
'"Giornale di Mat., 26, 1888, 86-95. 
227aNieuw Archief voor Wisk., 16, 1889, 209-214. 
=28Zeitschrift Math. Phys., 35, 1890, 382-i. 
22»Math. Annalen, 36, 1890, 304-315. ^'OBeitrage zur Zahlentheorie, 1891, 187. 
"iLucas, Th^orie des nombres, 1891, 411-2. Proof by H. W. Curjel, Math. Quest. Educ. 
Times, 57, 1892, 113. 
