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nuous functions of e; the remaining ones are continuous and of less 
consequence. 
A direct evaluation of the discontinuous term is not to be thought 
of; if however we suppose given beforehand the number of the prime 
numbers less than ¢, we can eliminate the discontinuous term and 
we are enabled, as will be shown subsequently, to arrive at rather 
close approximations of other more or less symmetric functions of 
these prime numbers. 
In order to obtain the formulae we have in view, we must apply 
RIEMANN’s method to the discontinuous integral 
1 atin 
Af c ‘ 
Gs (ce) = 5 f — log S (e + 5) de, 
2 ni z 
a—in 
the path of integration being a straight line parallel to the imaginary 
axis and on the positive side of it. 
By EureRr's equation we have, supposing z-+s > 1, 
n= 1 
_—_— 
log 6 (2-8) = S— Ep 
and by inserting this value in the integral we find 
n=o | 
G,()= = — = prs. 
n—=1 % pce 
On the other hand we may express ¢(2-+s) as an infinite 
product, that is we may write 
ee 
log £ (z + 8) = — log 2 — log (2 He — 1) +=, (€ + log 7) + 
z+s emai | ze+s 2-- 8) 
ikl reer) Meat See. | 
or by subtracting at both sides log ¢ (s), 
log £ (2 + 8) = log £ (s) — lag (1 — =) Hp (CH log) + 
S 
En (ne jk Zl fog (1 tie. 
nel 
where in the series 
