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.

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

${\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}.}${\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,

${\displaystyle \operatorname {li} (x)=\lim _{\varepsilon \to 0+}\left(\int _{0}^{1-\varepsilon }{\frac {dt}{\ln t}}+\int _{1+\varepsilon }^{x}{\frac {dt}{\ln t}}\right).}${\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.

The offset logarithmic integral or Eulerian logarithmic integral is defined as

${\displaystyle \operatorname {Li} (x)=\int _{2}^{x}{\frac {dt}{\ln t}}=\operatorname {li} (x)-\operatorname {li} (2).}${\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,

${\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=\operatorname {Li} (x)+\operatorname {li} (2).}${\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=\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... OEIS: A070769 ; this number is known as the Ramanujan–Soldner constant.

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

This is ${\displaystyle -(\Gamma (0,-\ln 2)+i,円\pi )}${\displaystyle -(\Gamma (0,-\ln 2)+i,円\pi )} where ${\displaystyle \Gamma (a,x)}${\displaystyle \Gamma (a,x)} is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.

The function li(x) is related to the exponential integral Ei(x) via the equation

${\displaystyle \operatorname {li} (x)={\hbox{Ei}}(\ln x),}${\displaystyle \operatorname {li} (x)={\hbox{Ei}}(\ln x),}

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

${\displaystyle \operatorname {li} (e^{u})={\hbox{Ei}}(u)=\gamma +\ln |u|+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\text{ for }}u\neq 0,,円}${\displaystyle \operatorname {li} (e^{u})={\hbox{Ei}}(u)=\gamma +\ln |u|+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\text{ for }}u\neq 0,,円}

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

${\displaystyle \operatorname {li} (e^{u})={\hbox{Ei}}(u)=\gamma +\ln u+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\text{ for }}u\neq 0,,円}${\displaystyle \operatorname {li} (e^{u})={\hbox{Ei}}(u)=\gamma +\ln u+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\text{ for }}u\neq 0,,円}

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

${\displaystyle \operatorname {li} (x)=\gamma +\ln |\ln x|+{\sqrt {x}}\sum _{n=1}^{\infty }\left({\frac {(-1)^{n-1}(\ln x)^{n}}{n!,2円^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}\right).}${\displaystyle \operatorname {li} (x)=\gamma +\ln |\ln x|+{\sqrt {x}}\sum _{n=1}^{\infty }\left({\frac {(-1)^{n-1}(\ln x)^{n}}{n!,2円^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}\right).}

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

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

${\displaystyle \operatorname {li} (x)=O\left({\frac {x}{\ln x}}\right).}${\displaystyle \operatorname {li} (x)=O\left({\frac {x}{\ln x}}\right).}

where ${\displaystyle O}${\displaystyle O} is the big O notation. The full asymptotic expansion is

${\displaystyle \operatorname {li} (x)\sim {\frac {x}{\ln x}}\sum _{k=0}^{\infty }{\frac {k!}{(\ln x)^{k}}}}${\displaystyle \operatorname {li} (x)\sim {\frac {x}{\ln x}}\sum _{k=0}^{\infty }{\frac {k!}{(\ln x)^{k}}}}

or

${\displaystyle {\frac {\operatorname {li} (x)}{x/\ln x}}\sim 1+{\frac {1}{\ln x}}+{\frac {2}{(\ln x)^{2}}}+{\frac {6}{(\ln x)^{3}}}+\cdots .}${\displaystyle {\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:

${\displaystyle \operatorname {li} (x)-{\frac {x}{\ln x}}=O\left({\frac {x}{(\ln x)^{2}}}\right).}${\displaystyle \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:

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

for all ${\displaystyle \ln x\geq 11}${\displaystyle \ln x\geq 11}.

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:

${\displaystyle \pi (x)\sim \operatorname {li} (x)}${\displaystyle \pi (x)\sim \operatorname {li} (x)}

where ${\displaystyle \pi (x)}${\displaystyle \pi (x)} denotes the number of primes smaller than or equal to ${\displaystyle x}${\displaystyle x}.

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

${\displaystyle |\operatorname {li} (x)-\pi (x)|=O({\sqrt {x}}\log x)}${\displaystyle |\operatorname {li} (x)-\pi (x)|=O({\sqrt {x}}\log x)}

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

${\displaystyle |\operatorname {li} (x)-\pi (x)|=O(x^{1/2+a})}${\displaystyle |\operatorname {li} (x)-\pi (x)|=O(x^{1/2+a})} for any ${\displaystyle a>0}${\displaystyle a>0}.

For small ${\displaystyle x}${\displaystyle x}, ${\displaystyle \operatorname {li} (x)>\pi (x)}${\displaystyle \operatorname {li} (x)>\pi (x)} but the difference changes sign an infinite number of times as ${\displaystyle x}${\displaystyle x} increases, and the first time that this happens is somewhere between 10¹⁹ and 1.4×ばつ10³¹⁶.

• Jørgen Pedersen Gram
• Skewes' number
• List of integrals of logarithmic functions

• 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 .

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