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.

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

Its value is approximately μ ≈ 1.45136923488338105028396848589202744949303228… (sequence A070769 in the OEIS)

Since the logarithmic integral is defined by

${\displaystyle \mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}},}$

then using ${\displaystyle \mathrm {li} (\mu )=0,}$ we have

${\displaystyle \mathrm {li} (x)\;=\;\mathrm {li} (x)-\mathrm {li} (\mu )=\int _{0}^{x}{\frac {dt}{\ln t}}-\int _{0}^{\mu }{\frac {dt}{\ln t}}=\int _{\mu }^{x}{\frac {dt}{\ln t}},}$

thus easing calculation for numbers greater than μ. Also, since the exponential integral function satisfies the equation

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

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… (sequence A091723 in the OEIS)