Tuples of polynomials over finite fields with pairwise coprimality conditions

Let q be a prime power. We estimate the number of tuples of degree bounded monic polynomials (Q1, . . . , Qv) ∈ (Fq[z])v that satisfy given pairwise coprimality conditions. We show how this generalises from monic polynomials in finite fields to Dedekind domains with a finite norm

On the distribution (mod 1) of the normalized zeros of the Riemann Zeta-function

We consider the problem whether the ordinates of the non-trivial zeros of ζ(s) are uniformly distributed modulo the Gram points, or equivalently, if the normalized zeros (xn) are uniformly distributed modulo 1. Odlyzko conjectured this to be true. This is far from being proved, even assuming the Riemann hypothesis (RH, for short). Applying the Piatetski-Shapiro 11/12 Theorem we are able to show that, for 0 < κ < 6/5, the mean value 1 N P n≤N exp(2πiκxn) tends to zero. The case κ = 1 is especially interesting. In this case the Prime Number Theorem is sufficient to prove that the mean value is 0, but the rate of convergence is slower than for other values of κ. Also the case κ = 1 seems to contradict the behavior of the first two million zeros of ζ(s). We make an effort not to use the RH. So our Theorems are absolute. We also put forward the interesting question: will the uniform distribution of the normalized zeros be compatible with the GUE hypothesis? Let ρ = 1 2 + iα run through the complex zeros of zeta. We do not assume the RH so that α may be complex. For 0 < κ < 6 5 we prove that lim T→∞ 1 N(T) X 0<Re α≤T e 2iκϑ(α) = 0 where ϑ(t) is the phase of ζ( 1 2 + it) = e −iϑ(t)Z(t).Ministerio de Economía y Competitivida

Asymptotics of Keiper-Li coefficients

We show that the Riemann Hypothesis is equivalent to the assertion (ym)∈ℓ2 where ymym is defined by λm=1/2(logm+γ−log(2π)−1)+ym, and mλm represents the numbers in Xian-Jin Li's criterion. This confirms and further sharpens a conjecture of J. B. Keiper. We also present some other hypotheses equivalent to the Riemann Hypothesis.Ministerio de Educación y Cienci

The problem of the notation for numerable ordinal numbers

Assuming the existence of inaccesible cardinal numbers it is proved that there is not a notation, for each numerable ordinal number, satisfying the conditions imposed by N. Cuesta Dutari in his book La matemática del orden (1959)

A test for the Riemann hypotesis

We prove that the Riemann Hypothesis holds if and only if I = Z +∞ 1 ˘ Π(x) − Li(x) ¯2 x −2 dx < +∞ with I = J, where J is some definite, computable real number (1.266 < J < 1.273). This provides us with a numerical test for the Riemann Hypothesis. The main interest of our test lies in the fact that it can also supply a goal. Namely, having computed J(a) := R a 1 ˘ Π(x) − Li(x) ¯2 x −2 dx < J for a number of values of a = an, we can estimate a value a for which, within our precision, we will have J(a) ≈ J.Ministerio de Ciencia e Innovació