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
l i ( x ) = ∫ 0 x d t ln t , {\displaystyle \mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}},}
then using l i ( μ ) = 0 , {\displaystyle \mathrm {li} (\mu )=0,} we have
l i ( x ) = l i ( x ) − l i ( μ ) = ∫ 0 x d t ln t − ∫ 0 μ d t ln t = ∫ μ x d t ln t , {\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
l i ( x ) = E i ( ln x ) , {\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)
External links Weisstein, Eric W. "Soldner's Constant". MathWorld.