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
or, integrally represented
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
li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151… A069284
The asymptotic behavior for x → ∞ is
This gives the following more accurate asymptotic behaviour:
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. (December 1972) . "Chapter 5". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 (10 ed.). New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 228. ISBN 978-0-486-61272-0. LCCN 64-60036. 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, MR 2723248