430 
History of the Theory of Numbers. 
[Chap. XVIII 
N, 
P. de Mond^sir-^^ wrote Np for the number of multiples of the prime p 
which are < 2.V and divisible by no prime < p. Then the number of primes 
<2.V is N—1iNp-\-n-\-l, where n is the number of primes <y/2N. Also, 
.p\ Lapj LahpA ' 
where a, h,. . . are the primes <p. By this modification of Legendre's 
formula, he computed the number 78490 of primes under one million. 
*L. Lorenz-^^ discussed the number of primes under a given limit. 
Paolo Paci"'' proved that the number of integers ^n divisible by a 
prime <\/ri is 
where r, s,. . . range over all the H primes 2, 3, . . . , p less than \/n. Thus 
there are n — \ —N-\-H primes from 1 to n. The approximate value of N is 
</>(2-3...p)" 
y r rs } y 2-3... p J 
Vs"^'J '\S 2-3... p 
K. E. Hoffmann--^ denoted by N the number of primes <m, by X the 
number of distinct prime factors of tw, by ^i the number of composite integers 
<m and prime to m. Evidently N = 4>{M) — ii-\-\. To find A'' it suffices 
to determine ix. To that end he would count the products <mhy twos, 
by threes, etc. (with repetitions) of the primes not dividing m. 
J. P. Gram-" proved that the number of powers of primes ^n is 
[Cf. Bougaief.^^^] Of the two proofs, one is by inversion from 
E. Cesaro^^ considered the number x of primes ^qn and >n, where 
g is a fixed prime. Let coi, . . . , co, be the primes ^ n other than 1 and q. 
Let5*^n<g*+\ Then 
Let Ir,, be the number of the [qn/{o3i . . .co,)] which give the remainder r when 
divided by q. Set t, = 'Ljlj,,. Then 
x=(k+l)q-{k + 2)-U + t2-h+ . . .. 
"'Assoc, frang. av. sc, 6, 1877, 77-92. Nouv. Corresp. Math., 6, 1880, 256. 
«»Tidsskr. for Math., Kjobenhavn, (4), 2, 1878, 1-3. 
""Sul numero de numeri primi inferiori ad un dato numero, Parma, 1879, 10 pp. 
»"Archiv Math. Phys., 64, 1879, 333-6. 
«K. Danake Vidensk. Selskabs. Skrifter, (6), 2, 1881-6, 183-288; r6sum6 in French, 289-308. 
See pp. 220-8, 296-8. 
»M6m. Soc. Sc. Li^e, (2), 10,«1883, 287-8. 
