# 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 .

## Integral representation

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 has to be interpreted as a *Cauchy principal 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.

## Special values

The function li(*x*) has a single positive zero; it occurs at *x* ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... OEIS: A070769; this number is known as the Ramanujan–Soldner constant.

−Li(0) = li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... OEIS: A069284

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

## Series representation

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 ... OEIS: A001620 is the Euler–Mascheroni constant. A more rapidly convergent series due to Ramanujan [1] is

## Asymptotic expansion

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 .

## Number theoretic significance

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 also

## References

- Weisstein, Eric W. "Logarithmic Integral".
*MathWorld*. - Abramowitz and Stegun, p. 230, 5.1.20

- 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