Ramanujan–Soldner constant explained

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 ≈ 1.45136923488338105028396848589202744949303228…

Since the logarithmic integral is defined by

li (x) = \int ^{x} _{0} (dt)/(ln t) ,

then using

li (\mu) = 0 , we haveli (x) = li (x) - li (\mu) = \int ^{x} _{0} (dt)/(ln t) - \int ^{\mu} _{0} (dt)/(ln t) = \int ^{x} _{\mu} (dt)/(ln t) ,

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

li (x) = 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…