Explicit smoothed prime ideals theorems under GRH

Abstract

Let ${\psi }_{\mathbb{K}}$ be the Chebyshev function of a number field $\mathbb{K}$. Let ${\psi }_{\mathbb{K}}^{(1)}(x):={\int }_0^x{\psi }_{\mathbb{K}}(t)dt$ and ${\psi }_{\mathbb{K}}^{(2)}(x):=2{\int }_0^x{\psi }_{\mathbb{K}}^{(1)}(t)dt$. We prove under GRH explicit inequalities for the differences $|{\psi }_{\mathbb{K}}^{(1)}(x)-\frac{x^2}{2}|$ and $|{\psi }_{\mathbb{K}}^{(2)}(x)-\frac{x^3}{3}|$. We deduce an efficient algorithm for the computation of the residue of the Dedekind zeta function and a bound on small-norm prime ideals.

Some misprints corrected, stronger conclusion in Th. 1.1. This is the final version which will appear in Mathematics of Computation

References in corpus (2)

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

Cited by in corpus (7)

• An effective Chebotarev density theorem under GRH
• Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH
• Primes and prime ideals in short intervals
• Primes in explicit short intervals on RH
• Explicit Short Intervals for Primes in Arithmetic Progressions on GRH
• Zeros of Dedekind zeta functions under GRH
• Explicit versions of the prime ideal theorem for Dedekind zeta functions under GRH, II