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Fungrim entry: a5d65f

(RH)    (n=1λna ⁣(n)2<   where a ⁣(n)=log ⁣(n)2log ⁣(2π)+1γ2)\left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a\!\left(n\right)\right|}^{2} < \infty\; \text{ where } a\!\left(n\right) = \frac{\log\!\left(n\right)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right)
References:
  • https://doi.org/10.7169/facm/1317045228
TeX:
\left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a\!\left(n\right)\right|}^{2} < \infty\; \text{ where } a\!\left(n\right) = \frac{\log\!\left(n\right)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right)
Definitions:
Fungrim symbol Notation Short description
RiemannHypothesisRH\operatorname{RH} Riemann hypothesis
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Powab{a}^{b} Power
Absz\left|z\right| Absolute value
KeiperLiLambdaλn\lambda_{n} Keiper-Li coefficient
Infinity\infty Positive infinity
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
ConstPiπ\pi The constant pi (3.14...)
ConstGammaγ\gamma The constant gamma (0.577...)
Source code for this entry:
Entry(ID("a5d65f"),
    Formula(Equivalent(RiemannHypothesis, Where(Less(Sum(Pow(Abs(Sub(KeiperLiLambda(n), a(n))), 2), Tuple(n, 1, Infinity)), Infinity), Equal(a(n), Sub(Div(Log(n), 2), Div(Sub(Add(Log(Mul(2, ConstPi)), 1), ConstGamma), 2)))))),
    References("https://doi.org/10.7169/facm/1317045228"))

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2019-09-20 18:07:53.062439 UTC