Explicit zero density theorems for Dedekind zeta functions

This article studies the zeros of Dedekind zeta functions. In particular, we establish a smooth explicit formula for these zeros and we derive an effective version of the Deuring-Heilbronn phenomenon. In addition, we obtain an explicit bound for the number of zeros in a box.Comment: 24 page

Une region explicite sans zero pour la fonction Zeta de Riemann

The Riemann Zeta function $\zeta(s)$ never vanishes in the region : $$\Re s \ge 1- \frac1{5.70176 \log |\Im s|} \quad \quad (|\Im s| \ge 2).$$Comment: 32 page

Density results for the zeros of zeta applied to the error term in the prime number theorem

We improve the unconditional explicit bounds for the error term in the prime counting function $\psi(x)$. In particular, we prove that, for all $x>2$, we have $$\left| \psi(x)-x \right| < 9.22106 \, x \, (\log x)^{3/2} \exp(-0.8476836\sqrt{\log x}),$$ and that, for all $\log x \ge 3\,000$, $$\left| \psi(x)-x \right| < 4.47\cdot 10^{-15} x.$$ This compares to results of Platt \& Trudgian (2021) who obtained $4.51\cdot 10^{-13} x$. Our approach represents a significant refinement of ideas of Pintz which had been applied by Platt and Trudgian. Improvements are obtained by splitting the zeros into additional regions, carefully estimating all of the consequent terms, and a significant use of computational methods. Results concerning $\pi(x)$ will appear in a follow up work

Management of stage one and two-E gastric large B-cell lymphoma: chemotherapy alone or surgery followed by chemotherapy?

Management of localized primary gastric B lymphoma (PGL) remains controversial. The aim of this study is to compare two treatments: chemotherapy alone and surgery plus chemotherapy