Fungrim entry: 3a5eb6

\left|\zeta\!\left(s\right)\right| \lt 3 \left|\frac{1 + s}{1 - s}\right| {\left|\frac{1 + s}{2 \pi}\right|}^{\left( 1 + \eta - \operatorname{Re}\left(s\right) \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}\!\left(s\right) \le 1 + \eta

References:

H. Rademacher, Topics in analytic number theory, Springer, 1973. Equation 43.3.

TeX:

\left|\zeta\!\left(s\right)\right| \lt 3 \left|\frac{1 + s}{1 - s}\right| {\left|\frac{1 + s}{2 \pi}\right|}^{\left( 1 + \eta - \operatorname{Re}\left(s\right) \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}\!\left(s\right) \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
`ConstPi`	$\pi$	The constant pi (3.14...)
`Re`	$\operatorname{Re}\!\left(z\right)$	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"),
    Formula(Less(Abs(RiemannZeta(s)), Mul(Mul(Mul(3, Abs(Div(Add(1, s), Sub(1, s)))), Pow(Abs(Div(Add(1, s), Mul(2, ConstPi))), Div(Sub(Add(1, eta), Re(s)), 2))), RiemannZeta(Add(1, eta))))),
    Variables(s, eta),
    Assumptions(And(Element(s, CC), Element(eta, RR), Unequal(s, 1), Element(eta, OpenClosedInterval(0, Div(1, 2))), LessEqual(Neg(eta), Re(s), Add(1, eta)))),
    References("H. Rademacher, Topics in analytic number theory, Springer, 1973. Equation 43.3."))

Topics using this entry

Riemann zeta function