On the Missing Log Factor
Résumé
This is a written account of a talk with the same title given during the main Chaire Morlet conference. Its guideline is the elementarily proven bound |\sum _{n\leqslant x}\mu (n)/n|\leqslant 1. The trivial bound for the implied summation is \log x+ \operatorname {\mathrm {\mathcal {O}}}(1), while the Prime Number Theorem tells us that it is o(1). Our starting estimate thus lies in-between, a fact that we explore under different lights.