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 .
The logarithmic integral has an integral representation defined for all positive real numbers ≠1 by the definite integral
\operatorname{li}(x)=
| x | |
| \int | |
| 0 |
| dt | |
| lnt |
.
Here, denotes the natural logarithm. The function has a singularity at, and the integral for is interpreted as a Cauchy principal value,
\operatorname{li}(x)=\lim\varepsilon\left(
| 1-\varepsilon | |
| \int | |
| 0 |
| dt | |
| lnt |
+
| x | |
| \int | |
| 1+\varepsilon |
| dt | |
| lnt |
\right).
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.
The offset logarithmic integral or Eulerian logarithmic integral is defined as
\operatorname{Li}(x)=
| x | |
| \int | |
| 2 |
| dt | |
| lnt |
=\operatorname{li}(x)-\operatorname{li}(2).
As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.
Equivalently,
\operatorname{li}(x)=
| x | |
| \int | |
| 0 |
| dt | |
| lnt |
=\operatorname{Li}(x)+\operatorname{li}(2).
The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... ; this number is known as the Ramanujan–Soldner constant.
\operatorname{li}(Li-1(0))=li(2)
This is
-(\Gamma(0,-ln2)+i\pi)
\Gamma(a,x)
The function li(x) is related to the exponential integral Ei(x) via the equation
\operatorname{li}(x)=\hbox{Ei}(lnx),
which is valid for x > 0. This identity provides a series representation of li(x) as
\operatorname{li}(eu)=\hbox{Ei}(u)=\gamma+ln|u|+
| infty | |
| \sum | |
| n=1 |
{un\overn ⋅ n!} foru\ne0,
where γ ≈ 0.57721 56649 01532 ... is the Euler–Mascheroni constant. For the complex function the formula is
\operatorname{li}(eu)=\hbox{Ei}(u)=\gamma+lnu+
| infty | |
| \sum | |
| n=1 |
{un\overn ⋅ n!} foru\ne0,
(without taking the absolute value of u).A more rapidly convergent series by Ramanujan is
\operatorname{li}(x)= \gamma +ln|lnx| +\sqrt{x}
| infty | |
| \sum | |
| n=1 |
\left(
| (-1)n-1(lnx)n | |
| n!2n-1 |
| \lfloor(n-1)/2\rfloor | |
| \sum | |
| k=0 |
| 1 | |
| 2k+1 |
\right).
ln|lnu|
lnlnu.
The asymptotic behavior both for
x\toinfty
x\to0+
\operatorname{li}(x)=O\left(
| x | |
| lnx |
\right).
O
\operatorname{li}(x)\sim
| x | |
| lnx |
| infty | |
| \sum | |
| k=0 |
| k! | |
| (lnx)k |
| \operatorname{li | |
| (x)}{x/ln |
x}\sim1+
| 1 | |
| lnx |
+
| 2 | |
| (lnx)2 |
+
| 6 | |
| (lnx)3 |
+ … .
This gives the following more accurate asymptotic behaviour:
\operatorname{li}(x)-
| x | |
| lnx |
=O\left(
| x | |
| (lnx)2 |
\right).
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+ | 1 |
| lnx |
<\operatorname{li}(x)
| lnx | |
| x |
<1+
| 1 | + | |
| lnx |
| 3 | |
| (lnx)2 |
lnx\ge11
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:
\pi(x)\sim\operatorname{li}(x)
\pi(x)
x
Assuming the Riemann hypothesis, we get the even stronger:[1]
|\operatorname{li}(x)-\pi(x)|=O(\sqrt{x}logx)
In fact, the Riemann hypothesis is equivalent to the statement that:
|\operatorname{li}(x)-\pi(x)|=O(x1/2+a)
a>0
For small
x
\operatorname{li}(x)>\pi(x)
x
