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Ramanujan–Soldner constant

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Ramanujan–Soldner constant

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

l i ( x ) = 0 x d t ln t ,

we have

l i ( x ) = l i ( x ) l i ( μ ) 0 x d t ln t = 0 x d t ln t 0 μ d t ln t l i ( x ) = μ x d t ln t ,

thus easing calculation for positive integers. Also, since the exponential integral function satisfies the equation

l i ( x ) = E 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)

References

Ramanujan–Soldner constant Wikipedia