Abstract We prove that the Riemann zeta-function ζ ( σ + i t ) has no zeros in the region σ ≥ 1 − 1 / ( 5.573412 log | t | ) for | t | ≥ 2 . This represents the largest known zero-free region… Expand

Abstract We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. The… Expand

It is known that $|\zeta(1+ it)|\ll (\log t)^{2/3}$. This paper provides a new explicit estimate, viz.\ $|\zeta(1+ it)|\leq 3/4 \log t$, for $t\geq 3$. This gives the best upper bound on $|\zeta(1+… Expand