Explicit estimates for the Riemann zeta function close to the 1-line (Q6612191)

In this paper, several explicit upper bounds of order \(\frac{\log t}{\log\log t}\) for the logarithmic derivative of the Riemann zeta-function \(\zeta(s)\) and its reciprocal when \(\sigma\) is close to 1 are obtained. Explicit bounds of this order inside the critical strip are provided for the first time. Other estimates are improvements of the current bounds on \(\sigma=1\).\N\NAs examples of such new bounds, we state the results obtained by the authors.\N\NThe first of them deals with the case \(t \geq t_0 \geq e^e\) and \(\sigma \geq 1 -\frac{\log \log t}{ W \log t}\). Let, for \(t_0=e^e\), \(W=22\) and \(R_1=5471\), while, for \(t_0=500\), \(R_2=3438\). Then it is shown that \[ \bigg|\frac{\zeta'(\sigma+it)}{\zeta(\sigma+it)}\bigg|\leq R_1\frac{\log t}{\log\log t}, \quad \text{and} \quad \bigg|\frac{1}{\zeta(\sigma+it)}\bigg|\leq R_2\frac{\log t}{\log\log t}. \]\N\NThe second part of the results deals with the case \(\sigma\geq 1\). Then, the following bounds are obtained \[ \bigg|\frac{\zeta'(\sigma+it)}{\zeta(\sigma+it)}\bigg|\leq K_1\frac{\log t}{\log\log t}, \quad t \geq t_0, \quad \text{and} \quad K_1=113.3 \ \ \text{for} \ \ t_0=500, \] and \[ \bigg|\frac{1}{\zeta(\sigma+it)}\bigg|\leq K_2\frac{\log t}{\log\log t}, \quad t \geq t_0 \geq e^e, \quad \text{and} \quad K_1=107.7 \ \ \text{for} \ \ t_0=500. \] Other values of \(R_1,R_2,K_1,K_2,W,t_0\) are also computed and provided in separate tables.\N\NNote that to obtain the estimates above mentioned, the authors are restricted by the bound on \(|\zeta(s)|\) in the region \(\sigma \geq 1 -\frac{\omega_2(\log \log t)^2}{\log t}\).