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, occurring in the prime number theorem as an estimate of the number of prime numbers less than a given value.
Offset logarithmic integral
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.
This function is a very good approximation to the number of prime numbers less than x.
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
The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 ...; this number is known as the Ramanujan–Soldner constant.
li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151…
The asymptotic behavior for x → ∞ is
Note that, 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.
Number theoretic significance
where denotes the number of primes smaller than or equal to .
- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 5", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 228, ISBN 978-0486612720, MR 0167642.
- Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248