Ramanujan–Soldner constant

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…

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

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

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

External links

• Weisstein, Eric W. "Soldner's Constant". MathWorld.