Ramanujan–Soldner constant

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Ramanujan–Soldner constant as seen on the logarithmic integral function.

In mathematics, the Ramanujan–Soldner constant (also called the Soldner constant) is a mathematical constant defined as the unique positive zero of the logarithmic integral function. It is named after Srinivasa Ramanujan and Johann Georg von Soldner.

Its value is approximately μ ≈ 1.451369234883381050283968485892027449493… (sequence A070769 in OEIS)

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}(\mu)
 \int_0^x \frac{dt}{\ln t} = \int_0^x \frac{dt}{\ln t} - \int_0^{\mu} \frac{dt}{\ln t}
 \mathrm{li}(x) = \int_{\mu}^x \frac{dt}{\ln t},

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


the only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan–Soldner constant, whose value is approximately ln(μ) ≈ 0.372507410781366634461991866… (sequence A091723 in OEIS)

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