Fungrim entry: 7783f9

\left(\operatorname{RH}\right) \iff \left(\frac{1}{\pi} \int_{0}^{\infty} \log\!\left(\left|\frac{\zeta\!\left(\frac{1}{2} + i t\right)}{\zeta\!\left(\frac{1}{2}\right)}\right|\right) \frac{1}{{t}^{2}} \, dt = \frac{\pi}{8} + \frac{\gamma}{4} + \frac{\log\!\left(8 \pi\right)}{4} - 2\right)

References:

https://mathoverflow.net/q/279936

TeX:

\left(\operatorname{RH}\right) \iff \left(\frac{1}{\pi} \int_{0}^{\infty} \log\!\left(\left|\frac{\zeta\!\left(\frac{1}{2} + i t\right)}{\zeta\!\left(\frac{1}{2}\right)}\right|\right) \frac{1}{{t}^{2}} \, dt = \frac{\pi}{8} + \frac{\gamma}{4} + \frac{\log\!\left(8 \pi\right)}{4} - 2\right)

Definitions:

Fungrim symbol	Notation	Short description
`RiemannHypothesis`	$\operatorname{RH}$	Riemann hypothesis
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Log`	$\log(z)$	Natural logarithm
`Abs`	$\left\|z\right\|$	Absolute value
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`ConstGamma`	$\gamma$	The constant gamma (0.577...)

Source code for this entry:

Entry(ID("7783f9"),
    Formula(Equivalent(RiemannHypothesis, Equal(Mul(Div(1, Pi), Integral(Mul(Log(Abs(Div(RiemannZeta(Add(Div(1, 2), Mul(ConstI, t))), RiemannZeta(Div(1, 2))))), Div(1, Pow(t, 2))), For(t, 0, Infinity))), Sub(Add(Add(Div(Pi, 8), Div(ConstGamma, 4)), Div(Log(Mul(8, Pi)), 4)), 2)))),
    References("https://mathoverflow.net/q/279936"))

Topics using this entry

Riemann hypothesis