854 
Tu == QU. 
ROE 
2. Takené ps di a, > a; 
t, is then divisible by 
(pF ye, Al) er oh ee In pi; 
therefore by p*t!, so that in this case too, we have 
Tu = Ws 
F (u) = 0. 
oale Gy Pr derij 
we have then 
WP Pere Ae ele iy 
and from 
Tu =P’ To, w= p*q 
it follows that there are only two possibilities, viz. 
a) E (uy) st, Ben T, = 9 Aceh a) 
b) EH) 0; ©) == Ú, To, ag; Oy) — 0s 
hence in this case 
E(u) == TAO) 
4. Take e, 5 p—1; 
as t, is divisible by e, +1, consequently by a number <p, 1, is 
in this case unequal to g, hence 
fav) = 0: 
Now that it has been proved that the conditions stated are 
satisfied, we are allowed to apply the proposition and formula (1) 
gives at once the relation sought. 
Finally we observe: in application II some asy mptotical expressions 
have been written for 2,(«), oe) and o,(7), but Lanpau deduces 
still sharper formulae for these functions. He ‘proves') that for 
each positive integral value of q, constant numbers A7, by, and 
Cu,» are to be found, for which the relations 
gnl (log log x)® a 
Th (2) = av = = Aas = = = a O Sait 
a=! b=0 (log x)@ (log «)7 
q sae log log x)? x 
ne ee, (ac ) A 
eae (log x) (log a) 
and gnl log log a) ah 
On (x) = fe PEN COP Oe 
a=1 b=0 (log a)“ (log x)9 
1) E. Lanpau. Ueber die Verteilung der Zahlen, welche aus » Primfaktoren 
zusammengesetzt sind. [Nachrichten von der Königlichen Gesellschaft der Wissen- 
schaften zu Gétlingen. Math.-physikalische Klasse. (1911). pages 361—381). 
