arXiv subjects
Publications and source records attributed to Giuseppe Molteni.
In [BDyDF08] Belabas, Diaz y Diaz and Friedman show a way to determine, assuming the Generalized Riemann Hypothesis, a set of prime ideals that generate the class group of a number field. Their method is efficient because it produces a set of ideals that is smaller than earlier proved results. Here we show how to use their main result to algorithmically produce a bound that is lower than the one they prove.
On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.
Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedekind zeta function having imaginary part in $[T-a,T+a]$. We also prove a bound for the multiplicity of the zeros.
Let $ψ_\K$ be the Chebyshev function of a number field $\K$. Under GRH we prove an explicit upper bound for $|ψ_\K(x)-x|$ in terms of the degree and the discriminant of $\K$. The new bound improves significantly on previous known results.
Let $ψ_{\mathbb K}$ be the Chebyshev function of a number field $\mathbb K$. Let $ψ^{(1)}_{\mathbb K}(x):=\int_{0}^{x}ψ_{\mathbb K}(t)\,d t$ and $ψ^{(2)}_{\mathbb K}(x):=2\int_{0}^{x}ψ^{(1)}_{\mathbb K}(t)\,d t$. We prove under GRH explicit inequalities for the differences $|ψ^{(1)}_{\mathbb K}(x) - \tfrac{x^2}{2}|$ and $|ψ^{(2)}_{\mathbb K}(x) - \tfrac{x^3}{3}|$. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals.