Ramanujan-Soldner constant Did You Mean Ramanujan-Soldner?

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The Ramanujan-Soldner constant is a mathematical constant defined as the unique positive zero of the logarithmic integral function.

Its value is approximately λ ≈ 1.451369234883381050283968485892027449493w/.

Since the logarithmic integral is defined by

$\mathrm{Li}(x) = \int_0^x \frac{dt}{\ln t},$

we have

$\mathrm{Li}(x)\;=\;\mathrm{Li}(x) - \mathrm{Li}(\lambda)$

$\int_0^x \frac{dt}{\ln t} = \int_0^x \frac{dt}{\ln t} - \int_0^{\lambda} \frac{dt}{\ln t}$

$\mathrm{Li}(x) = \int_{\lambda}^x \frac{dt}{\ln t},$

thus easing calculation for positive integers. Also, since the exponential integral function satisfies the equation

$\mathrm{Li}(x)\;=\;\mathrm{Ei}(\ln{x})$ ,

the only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan-Soldner constant, whose value is approximately ln(λ) ≈ 0.372507410781366634461991866w/.

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