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Fungrim entry: 3a5eb6

ζ ⁣(s)<31+s1s1+s2π(1+ηRe(s))/2ζ ⁣(1+η)\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:sC  and  ηR  and  s1  and  η(0,12]  and  ηRe(s)1+ηs \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
Absz\left|z\right| Absolute value
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
Powab{a}^{b} Power
Piπ\pi The constant pi (3.14...)
ReRe(z)\operatorname{Re}(z) Real part
CCC\mathbb{C} Complex numbers
RRR\mathbb{R} Real numbers
OpenClosedInterval(a,b]\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, Pi))), Div(Sub(Add(1, eta), Re(s)), 2))), RiemannZeta(Add(1, eta))))),
    Variables(s, eta),
    Assumptions(And(Element(s, CC), Element(eta, RR), NotEqual(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."))

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2020-04-08 16:14:44.404316 UTC