Simple question: what is 'x' in your formula?...
Kind regards
Hello,
Could anyone please give me a proof using basic/elementary number theory and or calculus of the following:
The sum of the reciprocals of primes is approximately equal to log(log(x))
e.g 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... ~ log(log(x))
Thanks in advance
All right!... the procedure that follows is due to Leohard Euler and starts with the 'infinite product' that brings his name...
(1)
... where with 'p' we indicate the sequence of prime numbers. For n=1 the (1) becomes...
(2)
Taking logarithm on both sides of (2)...
(3)
... where is a positive constant and is for k prime and elsewhere. Euler had demonstrated before that...
(4)
... where is another positive constant and is , so that for n 'large enough' is ...
(5)
... i.e. is asymptotic to . Now observing (3) we conclude that is asymptotic to i.e. for n 'large enough' is...
(6)
... or more precisely is...
(7)
Kind regards
In order to well undestand please observe this figure...
... where are reported...
a) in blue the function...
... where...
, k prime, elsewhere...
b) in red the function...
c) in grey the function...
What Euler has demonstrated two and half centuries ago [and that i replied now...] is that...
... and nothing else...
Kind regards