# logarithmic integral

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##### 1: 6.2 Definitions and Interrelations
###### §6.2(i) Exponential and LogarithmicIntegrals
The logarithmic integral is defined by
6.2.8 $\operatorname{li}\left(x\right)=\pvint_{0}^{x}\frac{\,\mathrm{d}t}{\ln t}=% \operatorname{Ei}\left(\ln x\right),$ $x>1$.
6.2.16 $\operatorname{Chi}\left(z\right)=\gamma+\ln z+\int_{0}^{z}\frac{\cosh t-1}{t}% \,\mathrm{d}t.$
##### 2: 6.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the exponential integrals $\operatorname{Ei}\left(x\right)$, $E_{1}\left(z\right)$, and $\operatorname{Ein}\left(z\right)$; the logarithmic integral $\operatorname{li}\left(x\right)$; the sine integrals $\operatorname{Si}\left(z\right)$ and $\operatorname{si}\left(z\right)$; the cosine integrals $\operatorname{Ci}\left(z\right)$ and $\operatorname{Cin}\left(z\right)$. …
##### 3: 6.16 Mathematical Applications
###### §6.16(ii) Number-Theoretic Significance of $\operatorname{li}\left(x\right)$
If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta\left(s\right)$ have real part of $\tfrac{1}{2}$25.10(i)), then
6.16.5 $\operatorname{li}\left(x\right)-\pi(x)=O\left(\sqrt{x}\ln x\right),$ $x\to\infty$,
##### 4: 27.12 Asymptotic Formulas: Primes
27.12.5 $\left|\pi\left(x\right)-\operatorname{li}\left(x\right)\right|=O\left(x\exp% \left(-c(\ln x)^{1/2}\right)\right),$ $x\to\infty$.
For the logarithmic integral $\operatorname{li}\left(x\right)$ see (6.2.8). …
27.12.6 $\left|\pi\left(x\right)-\operatorname{li}\left(x\right)\right|=O\left(x\exp% \left(-d(\ln x)^{3/5}\,(\ln\ln x)^{-1/5}\right)\right).$
$\pi\left(x\right)-\operatorname{li}\left(x\right)$ changes sign infinitely often as $x\to\infty$; see Littlewood (1914), Bays and Hudson (2000). …
27.12.7 $\left|\pi\left(x\right)-\operatorname{li}\left(x\right)\right|<\frac{1}{8\pi}% \sqrt{x}\,\ln x.$
##### 5: 4.10 Integrals
###### §4.10(i) Logarithms
For $\operatorname{li}\left(x\right)$ see §6.2(i). … Extensive compendia of indefinite and definite integrals of logarithms and exponentials include Apelblat (1983, pp. 16–47), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 107–116), Gröbner and Hofreiter (1950, pp. 52–90), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.3, 1.6, 2.3, 2.6).
##### 7: 27.21 Tables
Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare $\pi\left(x\right),\ifrac{x}{\ln x}$, and $\operatorname{li}\left(x\right)$. …
##### 8: 6.3 Graphics
For a graph of $\operatorname{li}\left(x\right)$ see Figure 6.16.2. …
##### 9: 6.14 Integrals
6.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{si}\left(t\right)\,\mathrm{d}t=-\frac{1}% {a}\operatorname{arctan}a,$ $\Re a>0$.
##### 10: 6.12 Asymptotic Expansions
###### §6.12(i) Exponential and LogarithmicIntegrals
6.12.2 $\operatorname{Ei}\left(x\right)\sim\frac{e^{x}}{x}\left(1+\frac{1!}{x}+\frac{2% !}{x^{2}}+\frac{3!}{x^{3}}+\cdots\right),$ $x\to+\infty$.
For the function $\chi$ see §9.7(i). The asymptotic expansion of $\operatorname{li}\left(x\right)$ as $x\to\infty$ is obtainable from (6.2.8) and (6.12.2). …
6.12.7 $R_{n}^{(\mathrm{f})}(z)=(-1)^{n}\int_{0}^{\infty}\frac{e^{-zt}t^{2n}}{t^{2}+1}% \,\mathrm{d}t,$