### A conditional density theorem for the zeros of the Riemann zeta-function

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According to the well-known Nyman-Beurling criterion the Riemann hypothesis is equivalent to the possibility of approximating the characteristic function of the interval $\left(0,1\right]$ in mean square norm by linear combinations of the dilations of the fractional parts $\left\{1/ax\right\}$ for real $a$ greater than $1$. It was conjectured and established here that the statement remains true if the dilations are restricted to those where the $a$’s are positive integers. A constructive sequence of such approximations is given.

We prove an explicit bound for N(σ,T), the number of zeros of the Riemann zeta function satisfying ℜ𝔢 s ≥ σ and 0 ≤ ℑ𝔪 s ≤ T. This result provides a significant improvement to Rosser's bound for N(T) when used for estimating prime counting functions.

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for $\zeta \left(s\right)$. We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper “Remarques sur un beau rapport entre...