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Fungrim entry: cf70ce

(RH)    (0112t2(1+4t2)31/2log ⁣(ζ ⁣(σ+it))dσdt=π(3γ)32)\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
RiemannHypothesisRH\operatorname{RH} Riemann hypothesis
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Loglog(z)\log(z) Natural logarithm
Absz\left|z\right| Absolute value
RiemannZetaζ(s)\zeta(s) Riemann zeta function
ConstIii Imaginary unit
Infinity\infty Positive infinity
ConstPiπ\pi The constant pi (3.14...)
ConstGammaγ\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(ConstPi, Sub(3, ConstGamma)), 32)))),
    References("https://doi.org/10.1007/BF01056314"))

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2019-10-05 13:11:19.856591 UTC