( 603 ) 
time its argument becomes equal to the first, second, third,... power 
of a prime number. 
This suggests that we eliminate the discontinuous function between 
the two equations and merely retain the relation 
GO — log | £ (5) | — Liltje |G, O+ bog 2 — il = 
FS 
2sc cos (/? log c) 3 B 
> ee AS 
y+ i+ 
a (3° 
loge Le e2+s loge © 
Whatever may be here the values of the coefficients A and B, 
from what precedes we may infer that they are finite and rather 
1 
small, so that for es, and for tolerably large values of c the 
right-hand side tends rapidly to zero. 
Regarding it as a vanishing quantity we are led to conclude 
that the relation 
[EO = log LE | iest ers [6 (9) + log 2 — Li) 
furnishes approximatively the value of Gs(c) as soon as Go(e) be 
given, that is, as soon as we know how many prime numbers and 
powers of prime numbers are to be found among the integers not 
exceeding c. 
The last equation necessarily takes a slightly altered form when 
s is tending to unity. In that case we must make use of the 
expansions 
log | § (s) | = — log (s — 1) + (8s — 1) Pi (s — J), 
Li (c+!) = C + log log es—! + (s — 1) Py (s — 1), 
from which we have ultimately 
Zim [log | £ (8) | + Li (e—#1)] = C + log log o 
Hence the value of Gj (c) may be derived approximately from 
G eA Ces ring | ZE oel gi (c) + log 2 — Li OF 
Moreover it is evident that a relation similar to that between 
