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 prime number 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 $x$ .

Integral representation

The logarithmic integral has an integral representation defined for all positive real numbers $x$ ≠ 1 by the definite integral

li (x) = \int_{0}^{x} \frac{d t}{\ln t} .

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,

li (x) = lim_{ε \to 0 +} (\int_{0}^{1 - ε} \frac{d t}{\ln t} + \int_{1 + ε}^{x} \frac{d t}{\ln t}) .

However, the logarithmic integral can also be taken to be a meromorphic complex-valued function in the complex domain. In this case it is multi-valued with branch points at 0 and 1, and the values between 0 and 1 defined by the above integral are not compatible with the values beyond 1. The complex function is shown in the figure above. The values on the real axis beyond 1 are the same as defined above, but the values between 0 and 1 are offset by iπ so that the absolute value at 0 is π rather than zero. The complex function is also defined (but multi-valued) for numbers with negative real part, but on the negative real axis the values are not real.

Offset logarithmic integral

The offset logarithmic integral or Eulerian logarithmic integral is defined as

Li (x) = \int_{2}^{x} \frac{d t}{\ln t} = li (x) - li (2) .

As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.

Equivalently,

li (x) = \int_{0}^{x} \frac{d t}{\ln t} = Li (x) + li (2) .

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.

$li ({Li}^{- 1} (0)) = li (2)$ ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... OEIS: A069284

This is $- (Γ (0, - \ln 2) + i π)$ where $Γ (a, x)$ 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

li (x) = Ei (\ln x),

which is valid for x > 0. This identity provides a series representation of li(x) as

li (e^{u}) = Ei (u) = γ + \ln | u | + \sum_{n = 1}^{\infty} \frac{u^{n}}{n \cdot n!} for u \neq 0,

where γ ≈ 0.57721 56649 01532 ... OEIS: A001620 is the Euler–Mascheroni constant. For the complex function the formula is

li (e^{u}) = Ei (u) = γ + \ln u + \sum_{n = 1}^{\infty} \frac{u^{n}}{n \cdot n!} for u \neq 0,

(without taking the absolute value of u). A more rapidly convergent series by Ramanujan^[1] is

li (x) = γ + \ln | \ln x | + \sqrt{x} \sum_{n = 1}^{\infty} (\frac{(- 1)^{n - 1} (\ln x)^{n}}{n! 2^{n - 1}} \sum_{k = 0}^{⌊ (n - 1) / 2 ⌋} \frac{1}{2 k + 1}) .

Again, for the meromorphic complex function the term $\ln | \ln u |$ must be replaced by $\ln \ln u .$

Asymptotic expansion

The asymptotic behavior both for $x \to \infty$ and for $x \to 0^{+}$ is

li (x) = O (\frac{x}{\ln x}) .

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

li (x) \sim \frac{x}{\ln x} \sum_{k = 0}^{\infty} \frac{k!}{(\ln x)^{k}}

or

\frac{li (x)}{x / \ln x} \sim 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^{2}} + \frac{6}{(\ln x)^{3}} + \dots .

This gives the following more accurate asymptotic behaviour:

li (x) - \frac{x}{\ln x} = O (\frac{x}{(\ln x)^{2}}) .

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:

1 + \frac{1}{\ln x} < li (x) \frac{\ln x}{x} < 1 + \frac{1}{\ln x} + \frac{3}{(\ln x)^{2}}

for all $\ln x \geq 11$ .

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:

π (x) \sim li (x)

where $π (x)$ is the prime-counting function giving the number of primes less than or equal to $x .$

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

| li (x) - π (x) | = O (\sqrt{x} \log x)

In fact, the Riemann hypothesis is equivalent to the statement that:

| li (x) - π (x) | = O (x^{1 / 2 + a})

for any

a > 0

.

For small $x$ , $li (x) > π (x)$ but the difference changes sign an infinite number of times as $x$ increases; the first time that this happens is believed to be close to the known upper bound of 1.3984×10³¹⁶,^[3]^[4] although the only proved lower bound is 10¹⁹.

References

↑ Weisstein, Eric W. "Logarithmic Integral". MathWorld.
↑ Abramowitz and Stegun, p. 230, 5.1.20
↑ Chao, Kuok Fai; Plymen, Roger (May 2010) [23 March 2009]. "A new bound for the smallest x with π(x) > li(x)". International Journal of Number Theory. 6 (3): 681–690. arXiv:math/0509312. doi:10.1142/S1793042110003125. we prove that there exists x in the interval [exp(727.951858), exp(727.952178)] for which π(x) − li(x) > 3.2×10¹⁵¹.
↑ Demichel, Patrick (5 April 2005). "The prime counting function and related subjects" (PDF). version 3.1415. Archived from the original (PDF) on 2007-02-04. In this paper I will show the heuristic that I use to demonstrate that the smallest value for x that satisfies the equation Li(x) < π(x) is the value 1.397162914e316

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 5". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 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.

[1] Weisstein, Eric W. "Logarithmic Integral". MathWorld.

[2] Abramowitz and Stegun, p. 230, 5.1.20

[3] Chao, Kuok Fai; Plymen, Roger (May 2010) [23 March 2009]. "A new bound for the smallest x with π(x) > li(x)". International Journal of Number Theory. 6 (3): 681–690. arXiv:math/0509312. doi:10.1142/S1793042110003125. we prove that there exists x in the interval [exp(727.951858), exp(727.952178)] for which π(x) − li(x) > 3.2×10¹⁵¹.

[4] Demichel, Patrick (5 April 2005). "The prime counting function and related subjects" (PDF). version 3.1415. Archived from the original (PDF) on 2007-02-04. In this paper I will show the heuristic that I use to demonstrate that the smallest value for x that satisfies the equation Li(x) < π(x) is the value 1.397162914e316

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[2]

[3]

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