Loic Andre Henri Grenie

Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH

We have recently proved several explicit versions of the prime ideal theorem under GRH. Here we further explore the method, in order to deduce its strongest consequence for the case where x diverges.

Explicit smoothed prime ideals theorems under GRH

Let ψK be the Chebyshev function of a number field K. Let ψ K (x) := ∫ x 0 ψK(t) dt and ψ (2) K (x) := 2 ∫ x 0 ψ (1) K (t) dt. We prove under GRH (Generalized Riemann Hypothesis) explicit inequalities for the differences |ψ K (x) − x 2 | and |ψ K (x) − x 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. Math. Comp. 85(300), 1875–1899 (2016). Electronically published on October 6, 2015. DOI: http://dx.doi.org/10.1090/mcom3039

Primes in explicit short intervals on RH

On the assumption of the Riemann hypothesis, we give explicit upper bounds on the difference between consecutive prime numbers.

Zeros of Dedekind zeta functions under GRH

Assuming GRH, we prove an explicit upper bound for the number of zeros of a Dedeking zeta function having imaginary part in [T-a, T+a]. We also prove a bound for the multiplicity of the zeros.

Explicit bounds for generators of the class group

Assuming Generalized Riemanns Hypothesis, Bach proved that the class group Cl_K of a number field K may be generated using prime ideals whose norm is bounded by 12 log^2 Delta_K, and by (4+o(1))log^2 Delta_K asymptotically, where Delta_K is the absolute value of the discriminant of K. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates Cl_K and which performs better than Bachs bound in computations, but which is asymptotically worse. In this paper we show that Cl_K is generated by prime ideals whose norm is bounded by the minimum of 4.01 log^2 Delta_K, 4(1+(2 pi e^gamma)^(-n_K))^2 log^2 Delta_K and 4(log Delta_K + log log Delta_K - (gamma+log 2\pi)n_K+1+(n_K+1)(log(7 log Delta_K)/log Delta_K)^2. Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedmans algorithms, confirming that it has size of the order of (\log Delta_K/log log Delta_K)^2. In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log^2 Delta_K except for 7 out of the 31292 fields we tested.