Fungrim entry: a5d65f

\left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a(n)\right|}^{2} < \infty\; \text{ where } a(n) = \frac{\log(n)}{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(n)\right|}^{2} < \infty\; \text{ where } a(n) = \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right)

Definitions:

Fungrim symbol	Notation	Short description
`RiemannHypothesis`	$\operatorname{RH}$	Riemann hypothesis
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Abs`	$\left\|z\right\|$	Absolute value
`KeiperLiLambda`	$\lambda_{n}$	Keiper-Li coefficient
`Infinity`	$\infty$	Positive infinity
`Log`	$\log(z)$	Natural logarithm
`Pi`	$\pi$	The constant pi (3.14...)
`ConstGamma`	$\gamma$	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), For(n, 1, Infinity)), Infinity), Equal(a(n), Sub(Div(Log(n), 2), Div(Sub(Add(Log(Mul(2, Pi)), 1), ConstGamma), 2)))))),
    References("https://doi.org/10.7169/facm/1317045228"))

Fungrim entry: a5d65f

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