The next six exercises establish some estimates for the size of π(x), the number of primes...

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Use Exercises 1 and 2 to show that there are positive constants c1 and c2such that

for all x ≥ 2. (Compare this to the strong statement given in the prime number theorem, stated as Theorem 3.4 in Section 3.2.)

Exercise 1

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Use Exercise A to show that

Exercise A

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Let p be a prime and let n be a positive integer. Show that p divides  exactly

times, where t = [logp 2n]Conclude that if pr divides , then pr ≤2n.

Exercise 2

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Use Exercise 3 to show that

Exercise 3

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Use Exercises 4 and 5 to show that

Exercise 4

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Use Exercise I to show that

Exercise I

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Let p be a prime and let n be a positive integer. Show that p divides  exactly

times, where t = [logp 2n]Conclude that if pr divides , then pr ≤2n.

Exercise 5

The next six exercises establish some estimates for the size of π(x), the number of primes less than or equal to x. These results were originally proved in the nineteenth century by Chebyshev.

Show that the product of all primes between n and 2n is between  aud nπ(2n)−π(n). (Hint:Use the fact that every prime between n and 2n divides (2n)! but not (n!)2.)