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Fungrim entry: 3a5eb6

ζ(s)<31+s1s1+s2π(1+ηRe(s))/2ζ ⁣(1+η)\left|\zeta(s)\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:sCandηRands1andη(0,12]andηRe(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(s)\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(s) Riemann zeta function
Powab{a}^{b} Power
ConstPiπ\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, 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."))

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2019-10-05 13:11:19.856591 UTC