New Proofs of the Equivalent Statement of the Dirichlet Eta Function and of the Riemann Hypothesis
AZIZ ARBAI *
ENSIT - Ecole des Nouvelles Sciences d’ingénierie - Le Laboratoire Systémes, Contrôle et Décision (LSCD), Tanger, Morocco.
*Author to whom correspondence should be addressed.
Abstract
We will present two new proofs (and some of our old results) for the “Dirichlet eta” function \(S(s)=\sum_{n \geq 1} \frac{(-1)^n}{n^s}\) which would lead us to announce some new conjectures equivalent to that of the Riemann hypothesis.
Conclusion: The first conjecture announced: In the band s (s = r+ ic) a complex such that its real part is strictly between 0 and 1 (0 < r < 1), we have the real part of the Dirchlet function (S(s)) can only be zero in the straight line "the real part of s is equal to 0.5" (r = 0,5). The second conjecture informs us about what the zeros can be in the straight line r = 0.5.
Keywords: Riemann zeta-function, hardy’s functional equation, adherent poi, complex numbe