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)