# Ramanujan–Soldner constant

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

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… (sequence A070769 in the OEIS)

Since the logarithmic integral is defined by

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

we have

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

thus easing calculation for positive integers. 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)