FIRST SECTION

This adventure starts with the following formula, obtained by me and perhaps by others and valid for all complex numbers except at 0 and the negative reals.

            (1)

From now on in this section we suppose that the Riemann Hypothesis is true, the  will be the imaginary parts of the non-trivial zeros of the zeta function.

So we can rewrite formula (1) in the form

            (2)

We have the following asimptotic behavior as

The following inequality holds for real numbers greater than 1, (due to its second term)

And so, for real numbers greater than 1 we have

I believe that the inequality above is equivalent to the Riemann Hypothesis.

From (2), and the substitution of x by exp(ix), we can get

            (3)

Example 1: Integrating f(x) we get g(x).

            (4)

Example 2: Differentiating (3) at x=0 we obtain

            (5)

Example 3: From (3) we obtain

            (6)

The substitution  , leads to

            (7)

SECOND SECTION

Now I decided to study experimentally the following sum over prime numbers

and I observed that

I believe that

and also that this inequality is equivalent to the Riemann Hypothesis.

NOTE

The following result is known