Special function defined by an integral
Plot of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D 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 {\displaystyle x} .
Logarithmic integral function plot
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).}
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 {\text{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 \left(0,-\ln 2\right)+i\,\pi )} where Γ ( a , x ) {\displaystyle \Gamma \left(a,x\right)} 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 {\hbox{li}}(x)={\hbox{Ei}}(\ln x),\,\!} which is valid for x > 0. This identity provides a series representation of li(x ) as
li ( e u ) = Ei ( u ) = γ + ln | u | + ∑ n = 1 ∞ u n n ⋅ n ! for u ≠ 0 , {\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\;,} where γ ≈ 0.57721 56649 01532 ... OEIS: A001620 is the Euler–Mascheroni constant . A more rapidly convergent series by Ramanujan [1] is
li ( x ) = γ + ln | ln x | + x ∑ n = 1 ∞ ( ( − 1 ) n − 1 ( ln x ) n n ! 2 n − 1 ∑ k = 0 ⌊ ( n − 1 ) / 2 ⌋ 1 2 k + 1 ) . {\displaystyle \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).}
Asymptotic expansion The asymptotic behavior for x → ∞ is
li ( x ) = O ( x ln x ) . {\displaystyle \operatorname {li} (x)=O\left({\frac {x}{\ln x}}\right).} where O {\displaystyle O} is the big O notation . The full asymptotic expansion is
li ( x ) ∼ x ln x ∑ k = 0 ∞ k ! ( ln x ) k {\displaystyle \operatorname {li} (x)\sim {\frac {x}{\ln x}}\sum _{k=0}^{\infty }{\frac {k!}{(\ln x)^{k}}}} or
li ( x ) x / ln x ∼ 1 + 1 ln x + 2 ( ln x ) 2 + 6 ( ln x ) 3 + ⋯ . {\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:
li ( x ) − x ln x = O ( x ( ln x ) 2 ) . {\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:
1 + 1 ln x < li ( x ) ln x x < 1 + 1 ln x + 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 ln x ≥ 11 {\displaystyle \ln x\geq 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:
π ( x ) ∼ li ( x ) {\displaystyle \pi (x)\sim \operatorname {li} (x)} where π ( x ) {\displaystyle \pi (x)} denotes the number of primes smaller than or equal to x {\displaystyle x} .
Assuming the Riemann hypothesis , we get the even stronger:[2]
| li ( x ) − π ( x ) | = O ( 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:
| li ( x ) − π ( x ) | = O ( x 1 / 2 + a ) {\displaystyle |\operatorname {li} (x)-\pi (x)|=O(x^{1/2+a})} for any a > 0 {\displaystyle a>0} . For small x {\displaystyle x} , li ( x ) > π ( x ) {\displaystyle \operatorname {li} (x)>\pi (x)} but the difference changes sign an infinite number of times as x {\displaystyle x} increases, and the first time this happens is somewhere between 1019 and 1.4×10316 .
See also
References 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 .