Purchase Solution

Logarithmic integral: two forms

Not what you're looking for?

Ask Custom Question

Define the logarithmic integral li(x) as the integral of the function 1/(log t) from t = 2 to t = x, where x > 2 and "log" denotes the natural logarithm.

(a) Determine constants A and B such that li(x) can be expressed in the following two forms:

(i) li(x) = x/(log x) + A + g(x), where g(x) is the integral of the function 1/[(log t)^2] from t = 2 to t = x

(ii) li(x) = x/(log x) + x/[(log x)^2] + B + 2h(x), where h(x) is the integral of the function 1/[(log t)^3] from t = 2 to t = x

----------------------------------------------------------------------------------

(b) Use the form of li(x) given in (a)(i) above to prove that li(x) ~ x/(log x).

Deduce that the Prime Number Theorem can be expressed in the form pi(x) ~ li(x), where (for any real number x) pi(x) is the prime-counting function that gives the number of primes less than or equal to x.

Attachments
Purchase this Solution

Solution Summary

The two forms of the logarithmic integral are derived. Also, the first form is used to show that the logarithmic integral li(x) has the same asymptotic behavior as the function pi(x), where pi(x) is the prime-counting function.

Solution provided by:
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
Recent Feedback
  • "Thanks for your assistance. "
  • "Thank you. I understand now."
  • "Super - Thank You"
  • "Very clear. I appreciate your help. Thank you."
  • "Great. thank you so much!"
Purchase this Solution


Free BrainMass Quizzes
Probability Quiz

Some questions on probability

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.