Ramanujan–Soldner constant

File:Logarithmic Integral Function and Soldner Constant.png

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

In mathematics, the Ramanujan–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… (sequence A070769 in the OEIS)

Since the logarithmic integral is defined by

$\mathrm{l}\mathrm{i}(x)=\underset{0}{\overset{x}{\int }}\frac{dt}{\ln t},$

then using $\mathrm{l}\mathrm{i}(\mu )=0,$ we have

$\mathrm{l}\mathrm{i}(x)=\mathrm{l}\mathrm{i}(x)-\mathrm{l}\mathrm{i}(\mu )=\underset{0}{\overset{x}{\int }}\frac{dt}{\ln t}-\underset{0}{\overset{\mu }{\int }}\frac{dt}{\ln t}=\underset{\mu }{\overset{x}{\int }}\frac{dt}{\ln t},$

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

$\mathrm{l}\mathrm{i}(x)=\mathrm{E}\mathrm{i}(\ln x),$

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)

External links

Template:Asbox