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 ) = ∫ 0 x d t ln t . {\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\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 → 0 + ( ∫ 0 1 − ε d t ln t + ∫ 1 + ε x d t ln t ) . {\displaystyle \operatorname {li} (x)=\lim _{\varepsilon \to 0+}\left(\int _{0}^{1-\varepsilon }{\frac {dt}{\ln t}}+\int _{1+\varepsilon }^{x}{\frac {dt}{\ln t}}\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.
Offset logarithmic integral The offset logarithmic integral or Eulerian logarithmic integral is defined as
Li ( x ) = ∫ 2 x d t ln t = li ( x ) − li ( 2 ) . {\displaystyle \operatorname {Li} (x)=\int _{2}^{x}{\frac {dt}{\ln t}}=\operatorname {li} (x)-\operatorname {li} (2).}
As such, the integral representation has the advantage of avoiding the singularity in the domain of integration. Equivalently,
li ( x ) = ∫ 0 x d t ln t = Li ( x ) + li ( 2 ) . {\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=\operatorname {Li} (x)+\operatorname {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 ) {\displaystyle \operatorname {li} ({\text{Li}}^{-1}(0))={\text{li}}(2)} ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... OEIS: A069284 This is − ( Γ ( 0 , − ln 2 ) + i π ) {\displaystyle -(\Gamma (0,-\ln 2)+i\,\pi )} where Γ ( a , x ) {\displaystyle \Gamma (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 ) , {\displaystyle \operatorname {li} (x)={\hbox{Ei}}(\ln x),}
which is valid for x > 0. This identity provides a series representation of li(x) as
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