850 
This relation has been first proved by E. Lanpau; from the proposition 
& 
Se ee 
pa log « 
he deduced viz. in an elementary way these relations ')?) 
wv (log log or! 
(n—1)/ log « 
Tr (a) aw, 
@ (log log x)r- 1 
0, DU) Ne rr Se 
(D) (n—1)! log « 
and 
x (log log «)"—! 
One) = (n =) log x 
By using the deeper lying relation 
Lv Lv 
A OG 
psa loge é oO oe) 
he proves, also elementarily *) 
wx (log log x) | wv (log log x)"—2 
Jr (w) == : () sas ie ; 
(n—1)! log « log « 
wx (log log x)» « (log log x)"—2 
satay a= bt (len 9) 
(n— 1): / log Lv log t 
and 
w (log log zr! wx (log log oP? 
On (w) = EP: ON a HÚN 
(n—1)! log wv log av 
What I want to prove now is that these formulae are only 
special cases of the proposition. Take (wv) = 1, if wv be equal to a 
squareless number composed of ” prime factors and take /(w) = 0 
in the other cases; then we have 
x 
DE OS (| Nema, RE B 
U2 
if (uw) be equal to 1 or O according as the total number of prime 
factors of w is equal to ” or not, we have 
Ss Ee Sir 
2 
us 
and finally, by giving to Fw) ge value 1 or O, according as -the 
number of ron prime factors of w is equal to » or not, 
have 
1) E. LANDAU. Sur quelques problèmes relatifs a la distribution des nombres 
premiers. [Bulletin de la Société mathématique de France. Vol. 28 ie pg. 25—28}. 
#) KE. Lanpav. Handbuch I. p. 205—213. 
