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Concept
Logarithmic integral function
Summary
In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value .
The logarithmic integral has an integral representation defined for all positive real numbers x ≠ 1 by the definite integral
Here, ln denotes the natural logarithm. The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value,
The offset logarithmic integral or Eulerian logarithmic integral is defined as
As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.
Equivalently,
The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... ; this number is known as the Ramanujan–Soldner constant.
−Li(0) = li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151...
This is where is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.
The function li(x) is related to the exponential integral Ei(x) via the equation
which is valid for x > 0. This identity provides a series representation of li(x) as
where γ ≈ 0.57721 56649 01532 ... is the Euler–Mascheroni constant. A more rapidly convergent series by Ramanujan is
The asymptotic behavior for x → ∞ is
where is the big O notation. The full asymptotic expansion is
or
This gives the following more accurate asymptotic behaviour:
As an asymptotic expansion, this series is not convergent: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of x are employed. This expansion follows directly from the asymptotic expansion for the exponential integral.
This implies e.g. that we can bracket li as:
for all .
Official source
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