Harmonic numbers, harmonic series and zeta function

Abstract

This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the series 𝒨is(z)=∑n=1∞μ(n)nszn$\mathcal{M}{i_s}(z) = \sum\limits_{n = 1}^\infty {{{\mu (n)} \over {{n^s}}}{z^n}} $. A significant result is that the real part f of

Keywords

• harmonic numbers
• möbius function
• zeta functions
• explicit formulas
• sums of squares
• primary: 11e45
• 11a25
• 11a41
• 11r47