Harmonic numbers, harmonic series and zeta function

Abstract

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 𝒨 i s ( z ) = ∑ n = 1 ∞ μ ( n ) n s z n $\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 ∑ μ ( n ) n e 2 i n π θ $$\sum {{{\mu (n)} \over n}{e^{2in\pi \theta }}}$$ is an example of a non-trivial real-valued continuous function f on the real line which is 1-periodic, is not odd and has the property ∑ h = 1 n f ( h / k ) = 0 $\sum\nolimits_{h = 1}^n {f(h/k) = 0}$ for every positive integer k.