Chap. XIX] NUMERICAL INTEGRALS AND DERIVATIVES. 449 
The theorem Sn=iM(^)A = and other results on sums involving /i(n) 
play an important role 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,^^ and to the subsequent papers by A. Axer,^^ 
E. Landau/3 and J. F. Steffensen.^'' 
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, Theorie des nombres, Ann. fac. Toulouse, 4, 1890, 21. 
Borel and Drach, Introd. theorie des nombres, 1895, 24-6. 
E. Cahen, ;Sl6ments de la theorie des nombres, 1900, 346-350. 
E. Landau," 577-9. 
Numerical Integrals and Derivatives. 
N. V. Bougaief^^ (Bugaiev) called F{n) the numerical integral of /(n) if 
F(m) =2/(5), summed for all the divisors 5 of m, and called /(n) the numer- 
ical derivative function of F{n), denoted by DF(n) symbolically. 
Granting that there is, for every n, the development 
F{n) = a,[n]+a2y^j '^^Hjj 
+ 
where [x] is the largest integer ^x, then a^ is the numerical derivative of 
F{k) -F{k-1). He developed [n'^% [n'% etc. 
N. V. Bougaief,^^ after amplifying the preceding remarks, proved that 
S e{8)x{d) =yp{n), d(n)d(m) =d{nm) 
d6=n 
imply 
Writing D~^d{d) for '29(d), summed for the divisors d^ of n, we have 
D'^2x{md)=^2xi^)D''d{d), 
for any integer /x, positive or negative. There are formulas like 
"Handbuch. . .Verteilung der Primzahlen, II, 1909, 567-637, 676-96, 901-2. 
«Prace mat. fiz., Warsaw, 21, 1910, 65-95; Sitzungsber. Ak. Wiss. Wien (Math.), 120, 1911, 
II a, 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. Sbomik 
(Math. Soc. Moscow), 5, I, 1870-2, 1-63; 6, 1872-3, I, 133-180, 199-254, 309-360 
(reviewed in BuU. Sc. Math. Astr., 3, 1872, 200-2; 5, 1873, 296-8; 6, 1874, 314-6). 
R^8um6 by Bougaief, BuU. Sc. Math. Astr., 10, I, 1876, 13-32. 
