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Fungrim entry: 7783f9

(RH)    (1π0log ⁣(ζ ⁣(12+it)ζ ⁣(12))1t2dt=π8+γ4+log ⁣(8π)42)\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
RiemannHypothesisRH\operatorname{RH} Riemann hypothesis
ConstPiπ\pi The constant pi (3.14...)
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Loglog(z)\log(z) Natural logarithm
Absz\left|z\right| Absolute value
RiemannZetaζ(s)\zeta(s) Riemann zeta function
ConstIii Imaginary unit
Powab{a}^{b} Power
Infinity\infty Positive infinity
ConstGammaγ\gamma The constant gamma (0.577...)
Source code for this entry:
Entry(ID("7783f9"),
    Formula(Equivalent(RiemannHypothesis, Equal(Mul(Div(1, ConstPi), 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(ConstPi, 8), Div(ConstGamma, 4)), Div(Log(Mul(8, ConstPi)), 4)), 2)))),
    References("https://mathoverflow.net/q/279936"))

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2019-10-05 13:11:19.856591 UTC