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 .
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.
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 asymptotic behavior for x → ∞ 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 significance
where denotes the number of primes smaller than or equal to .
Assuming the Riemann hypothesis, we get the even stronger:
- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 5". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 228. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
- Temme, N. M. (2010), "Exponential, Logarithmic, Sine, and Cosine Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248