Ramanujan-Soldner constant

The Ramanujan-Soldner constant is a mathematical constant defined as the unique positive zero of the logarithmic integral function.

Related Topics:
Mathematical constant - Zero - Logarithmic integral function

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Its value is approximately λ ≈ 1.451369234883381050283968485892027449493...

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Since the logarithmic integral is defined by

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: mathrm{Li}(x) = int_0^x rac{dt}{ln t},

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

we have

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: mathrm{Li}(x);=;mathrm{Li}(x) - mathrm{Li}(lambda)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: int_0^x rac{dt}{ln t} = int_0^x rac{dt}{ln t} - int_0^{lambda} rac{dt}{ln t}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: mathrm{Li}(x) = int_{lambda}^x rac{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(λ) ≈ 0.372507410781366634461991866...

~ ~ ~ ~ ~ ~ ~ ~ ~ ~