Ramanujan–Soldner constant

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 ≈ 1.4513692348833810502839684858920274494933… (sequence A070769 in the OEIS)

Since the logarithmic integral is defined by

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

we have

{\mathrm {li}}(x)\;=\;{\mathrm {li}}(x)-{\mathrm {li}}(\mu )

\int _{0}^{x}{\frac {dt}{\ln t}}=\int _{0}^{x}{\frac {dt}{\ln t}}-\int _{0}^{{\mu }}{\frac {dt}{\ln t}}

{\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

{\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.

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