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CHAP. XIX]            NUMEBICAL INTEGHALS ANI> DERIVATIVES.                       449
The theorem Sn*1ju(n)/n = 0 and other results on sums involving p,(n) play an important r61e in the theory of the asymptotic distribution of primes. In accord with the plan of not entering into details on that topic (Ch. XVIII), the reader is referred for the former topic to the history and exposition by E. Landau,41 and to the subsequent papers by A. Axer,42 E. Landau,43 and J. F. Steffensen.44
Proofs of (2) or (3) are given in the following texts:
P. Bachmann, Die Lehre von der Kreistheilung, 1872, 8-11; Die Elemente der Zahlentheorie, 1892, 40-4; Grundlehren der Neueren Zahlentheorie, 1907, 26-9. T. J. Stieltjes, Throne des nombres, Ann. fac. Toulouse, 4, 1890, 21. Borel and Drach, Introd. the*orie des nombres, 1895, 24-6. E. Cahen, filaments de la the*orie des nombres, 1900, 346-350. E. Landau,41 577-9.
N. V. Bougaief65 (Bugaiev) called F(n) the numerical integral of f(n) if F(m) =Z/(5), summed for all the divisors 8 of m, and called f(ri) the numerical derivative function of F(ri), denoted by DF(ri) symbolically.
Granting that there is, for every n, the development
where [x] is the largest integer ^x, then ak is the numerical derivative of f(k) -F(k-l).   He developed [n1/2], [n1/3], etc.
N. V. Bougaief,66 after amplifying the preceding remarks, proved that
6(n) B(m) =0(nm)
Writing D~lQ(d) for S0(d), summed for the divisors d of n, we have
I«x(*)0W=Zx0)l>m for any integer //, positive or negative.   There are formulas like
"Handbuch. . .Verteilung der Primzahlen, II, 1909, 567-637, 676-96, 901-2.
«Prace mat. JBz., Warsaw, 21, 1910, 65-95; Sitzungeber. Ak. Wiss. Wien (Math.), 120, 1911,
Ila, 1253-98. «Sitzungsber. Ak. Wiss. Wien. (Math.), 120, 1911, Ila, 973-88; Rend. Circ. Mat. Palermo, 34,
1912, 121-31. "Analytiske Studier. . ., Diss., Kjobenhavn, 1912, 148 pp.   Fortschritte, 43, 262-3.    Extract
in Acta Math., 37, 1914, 75-112. "Journal de la Soc. Philomatique de Moscou, 5, 1871. "Theory of numerical derivatives, Moscow, 1870-3, 222 pp.   Extracts from  Mat. Sbornik
(Math. Soc. Moscow), 5, I, 1870-2, 1-63; 6, 1872-3,  I, 133-180, 199-254, 309-360
(reviewed in Bull. Sc. Math. Astr., 3, 1872, 200-2; 5, 1873, 296-8; 6, 1874, 314-6).
Eesume' by Bougaief, Bull. Sc. Math. Astr., 10, I, 1876, 13-32.