# Fungrim entry: 3a5eb6

$\left|\zeta\!\left(s\right)\right| < 3 \left|\frac{1 + s}{1 - s}\right| {\left|\frac{1 + s}{2 \pi}\right|}^{\left( 1 + \eta - \operatorname{Re}(s) \right) / 2} \zeta\!\left(1 + \eta\right)$
Assumptions:$s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{R} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \eta \in \left(0, \frac{1}{2}\right] \;\mathbin{\operatorname{and}}\; -\eta \le \operatorname{Re}(s) \le 1 + \eta$
References:
• H. Rademacher, Topics in analytic number theory, Springer, 1973. Equation 43.3.
TeX:
\left|\zeta\!\left(s\right)\right| < 3 \left|\frac{1 + s}{1 - s}\right| {\left|\frac{1 + s}{2 \pi}\right|}^{\left( 1 + \eta - \operatorname{Re}(s) \right) / 2} \zeta\!\left(1 + \eta\right)

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \eta \in \mathbb{R} \;\mathbin{\operatorname{and}}\; s \ne 1 \;\mathbin{\operatorname{and}}\; \eta \in \left(0, \frac{1}{2}\right] \;\mathbin{\operatorname{and}}\; -\eta \le \operatorname{Re}(s) \le 1 + \eta
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
RiemannZeta$\zeta\!\left(s\right)$ Riemann zeta function
Pow${a}^{b}$ Power
Pi$\pi$ The constant pi (3.14...)
Re$\operatorname{Re}(z)$ Real part
CC$\mathbb{C}$ Complex numbers
RR$\mathbb{R}$ Real numbers
OpenClosedInterval$\left(a, b\right]$ Open-closed interval
Source code for this entry:
Entry(ID("3a5eb6"),
References("H. Rademacher, Topics in analytic number theory, Springer, 1973. Equation 43.3."))