# 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

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2021-03-15 19:12:00.328586 UTC