Application 6. Let (pp) be the sequence of prime numbers. Then we
have the following theorems.
sent f(6) by the family of segments (In)n>1, where I; =]0, f (=)].
Now, from the above results, we deduce that
We obtain
limit f(0). Let (@n)n>o and (Bn)n>o0 be two real sequences such that
Property 11.16. For every natural number k > 1, we have
On the other hand, we know that Boy41 = 0 for every natural number
k > 1. To prove Ws+2%+41,s = 0, it is sufficient to show that the above
column has the following form
column it is sufficient to determine the values of the real numbers (@)
which verify
Then, the first column of 7)\") is given by
Property 11.18. For every k € N and s € N*, we have
To find the sequence of the real numbers (a;);>0, it is sufficient to de-
termine A € R, such that
The above result is not true, since the function f is not holomorphic
on a neighborhood of 0 by Lemma 11.22. Then, we reached to obtain
a contradiction and we deduce that the series (s,,) does not admit an
exact limit.
Finally, we find the standard Euler-Maclaurin formula [6] applied to
the zeta function ¢(s), where s is a natural number and s > 2. Then we
deduce that the matrix of the black magic w”) has a beautiful twelfth
property which is given as follows.