( 604 ) 
Gs(c) and G,(e) exists between two integrals Gs (ce) and G:(¢), and 
1 ; 
that for s >t and s > a we may write 
[Gs (0) — tag LEO | Lie |= 
= tt [64 (e) = log | EO | — Lil]. 
Lastly, we may remark that it is perfectly admissible to differen- 
tiate with respect to s the equation connecting Gs(¢) and Go (©). 
Remembering that 
d 
eo 5 Gs ol 
is equal to the logarithm of the least common multiple M(c) of all 
integers less than c, and that £(0):5(0) log 27, we find, by 
putting s equal to zero after the performance of the differentiation, 
log M (c) + log 20 — | — loge |G (c) + log 2 — Li 0 == 
ii 2¢3 [= cos (7 log c) 1E a B' 
logel ep as afs 3 log e 
Now although the second member increases as ec increases, it 
remains relatively small with respect to ¢ and log M (©); therefore we 
may expect the relation 
[lag M (0) + log 2 x — | = loge | (2) + bog 2 — Lie) | 
to furnish approximately the value of log M(e), 
The following test-cases abundantly show, that already for a c 
of moderate magnitude the approximation is very close. 
lL. ee = 7.389, Gale) = OI Gs (n= 018077. 
[Gs (c) — log 5 (8) — Li (e—*)] = 0.00052, 
e—? [Gy (c) — log § (2) — Li (e-?)] = 000054, 
IL. cae = 20.086, G, (ec) = 1.69330, Gy (c) = 0.48456. 
[Gs (©) — log £ (2) — Li (e-)] = 0.00089, 
e—° | Gy (c) — C — log 3] = 0.00088. 
nn 
