Logarithmic integral function

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 Siegel-Walfisz 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 .

Logarithmic integral function plot

Integral representationEdit

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,


Offset logarithmic integralEdit

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.

Special valuesEdit

The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... OEISA070769; this number is known as the Ramanujan–Soldner constant.

−Li(0) = li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... OEISA069284

This is   where   is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.

Series representationEdit

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 ... OEISA001620 is the Euler–Mascheroni constant. A more rapidly convergent series by Ramanujan [1] is


Asymptotic expansionEdit

The asymptotic behavior for x → ∞ is


where   is the big O notation. The full asymptotic expansion is




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  .

Number theoretic significanceEdit

The logarithmic integral is important in number theory, appearing in estimates of the number of prime numbers less than a given value. For example, the prime number theorem states that:


where   denotes the number of primes smaller than or equal to  .

Assuming the Riemann hypothesis, we get the even stronger:[2]


See alsoEdit


  1. ^ Weisstein, Eric W. "Logarithmic Integral". MathWorld.
  2. ^ Abramowitz and Stegun, p. 230, 5.1.20