Fungrim entry: cf70ce

\left(\operatorname{RH}\right) \iff \left(\int_{0}^{\infty} \frac{1 - 12 {t}^{2}}{{\left(1 + 4 {t}^{2}\right)}^{3}} \int_{1 / 2}^{\infty} \log\!\left(\left|\zeta\!\left(\sigma + i t\right)\right|\right) \, d\sigma \, dt = \frac{\pi \left(3 - \gamma\right)}{32}\right)

References:

https://doi.org/10.1007/BF01056314

TeX:

\left(\operatorname{RH}\right) \iff \left(\int_{0}^{\infty} \frac{1 - 12 {t}^{2}}{{\left(1 + 4 {t}^{2}\right)}^{3}} \int_{1 / 2}^{\infty} \log\!\left(\left|\zeta\!\left(\sigma + i t\right)\right|\right) \, d\sigma \, dt = \frac{\pi \left(3 - \gamma\right)}{32}\right)

Definitions:

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

Source code for this entry:

Entry(ID("cf70ce"),
    Formula(Equivalent(RiemannHypothesis, Equal(Integral(Mul(Div(Sub(1, Mul(12, Pow(t, 2))), Pow(Add(1, Mul(4, Pow(t, 2))), 3)), Integral(Log(Abs(RiemannZeta(Add(sigma, Mul(ConstI, t))))), For(sigma, Div(1, 2), Infinity))), For(t, 0, Infinity)), Div(Mul(Pi, Sub(3, ConstGamma)), 32)))),
    References("https://doi.org/10.1007/BF01056314"))

Topics using this entry

Riemann hypothesis