Chap. XVIII] NUMBER OF PRIMES. 429 
Number of Primes Between Assigned Limits. 
Formula (5) of Legendre in Ch. V implies that if 0, X, . . . are the primes 
= Vn, the mimber of primes ^ n and > Vn is one less (if unity be counted a 
prime) than 
statements or proofs of this result have been given by C. J. Hargreave,^^* 
E. de Jonquieres,206 R. Lipschitz,207 j. j. Sylvester,^^^ E. Catalan,^^^ F. Ro- 
gel,^^° J. Hammond^^^ with a modification, H. W. Curjel,^^^" S. Johnsen,^^^ 
and L. Kronecker.^^^ 
E. MeisseP^'* proved that if d{m) is the number of primes (including 
unity) ^m and if 
*(P.- . . . p."") = ( - 1)-+ •+"" ("■+"^+- •■+»»)! , 
rill. . .n-f 
1 
.$(l)^[f]+<i.(2)e[f] + ...+*Wflg]. 
E. MeisseP^^ wrote $(m, n) for Legendre's formula for the number of 
integers ^ m which are divisible by no one of the first n primes pi = 2, . . . , p„. 
Then .p^-, x 
<l>(m, n) =^{m, n — 1) — «l> ( — , n — 1 j • 
Let^(m)be the number of primes ^m. Setn+fx = d{\/m),n==d{^^/m). Then 
e{m)=^{m, n)+n(M+l)+ ^^^~'^^ -l- i d(^^), 
which is used to compute Birri) for m = ^-10^, A; = 1/2, 1, 10. 
MeisseP^^ applied his last formula to find ^(10^). 
Lionnet^^^'' stated that the number of primes between A and 2 A is 
<B{A). 
N. V. Bougaief2i7 obtained from Q{n) +^(n/2) +^(V3) + • • • =S[n/p], by 
inversion (Ch. XIX), 
^(^) =49 -^4^] +«4J^] - -45] +4Je] -<^c] + • ■ • . 
where a, h,. . . range over all primes. 
2«6Lond. Ed. Dub. Phil. Mag., (4), 8, 1854, 118-122. 
^o^Comptes Rendus Paris, 95, 1882, 1144, 1343; 96, 1883, 231. 
^oUbid., 95, 1882, 1344-6; 96, 1883, 58-61, 114-5, 327-9. 
2<>876id., 96, 1883, 463-5; Coll. Math. Papers, 4, p. 88. 
2°9Mem. Soc. Roy. Sc. de Liege, (2), 12, 1885, 119; Melanges Math., 1868, 133-5. 
JioArchiv Math. Phys., (2), 7, 1889, 381-8. ^iiMessenger Math., 20, 1890-1, 182. 
2"«Math. Quest. Educ. Times, 67, 1897, 27. 
"2Nyt Tidsskrift for Mat., Kjobenhavn, 15 A, 1904, 41^. 
2"Vorlesungen uber Zahlentheorie, I, 1901, 301-4. "ujour. fur Math., 48, 1854, 310-4. 
2i5Math. Ann., 2, 1870, 636-642. Outline in Mathews' Theory of Numbers, 273-8, and in G. 
Wertheim's Elemente der Zahlentheorie, 1887, 20-25. 
2i676id., 3, 1871, 523-5. Corrections, 21, 1883, 304. 
2i6aNouv. Ann. Math., 1872, 190. Cf. Landau, (4), 1, 1901, 281-2. 
21'Bull. sc. math, astr., 10, 1, 1876, 16. Mat. Sbomik (Math. Soc. Moscow), 6, 1872-3, I, 180. 
