# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC