Logarithmic integral function

The logarithmic integral has an integral representation defined for all positive real numbers  ≠ 1 by the definite integral : \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}. 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 \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 : \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, : \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... ; this number is known as the Ramanujan–Soldner constant. \operatorname{li}(\text{Li}^{-1}(0)) = \text{li}(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... This is -(\Gamma(0,-\ln 2) + i\,\pi) where \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 : \operatorname{li}(x)=\hbox{Ei}(\ln x) , which is valid for x > 0. This identity provides a series representation of li(x) as : \operatorname{li}(e^u) = \hbox{Ei}(u) = \gamma + \ln |u| + \sum_{n=1}^\infty {u^{n}\over n \cdot n!} \quad \text{ for } u \ne 0 \, , where γ ≈ 0.57721 56649 01532 ... is the Euler–Mascheroni constant. For the complex function the formula is : \operatorname{li}(e^u) = \hbox{Ei}(u) = \gamma + \ln u + \sum_{n=1}^\infty {u^{n}\over n \cdot n!} \quad \text{ for } u \ne 0 \, , (without taking the absolute value of u). A more rapidly convergent series by Ramanujan is : \operatorname{li}(x) = \gamma + \ln |\ln x| + \sqrt{x} \sum_{n=1}^\infty \left( \frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k+1} \right). 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 : \operatorname{li}(x) = O \left( \frac{x }{\ln x} \right) . where O is the big O notation. The full asymptotic expansion is : \operatorname{li}(x) \sim \frac{x}{\ln x} \sum_{k=0}^\infty \frac{k!}{(\ln x)^k} or : \frac{\operatorname{li}(x)}{x/\ln x} \sim 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \frac{6}{(\ln x)^3} + \cdots. This gives the following more accurate asymptotic behaviour: : \operatorname{li}(x) - \frac{x}{ \ln x} = O \left( \frac{x}{(\ln x)^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+\frac{1}{\ln x} for all \ln x \ge 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: : \pi(x)\sim\operatorname{li}(x) where \pi(x) denotes the number of primes smaller than or equal to x. Assuming the Riemann hypothesis, we get the even stronger: : |\operatorname{li}(x)-\pi(x)| = O(\sqrt{x}\log x) In fact, the Riemann hypothesis is equivalent to the statement that: : |\operatorname{li}(x)-\pi(x)| = O(x^{1/2+a}) for any a>0. For small x, \operatorname{li}(x)>\pi(x) but the difference changes sign an infinite number of times as x increases, and the first time that this happens is somewhere between 1019 and . == See also ==