CHAPTER 5
Number Theory and Modern Algebra
5.1 PRIME NUMBERS
The integer p is a prime integer, if it is divisible only by 1 and itself; otherwise, p is a composite integer. Although there are infinitely many primes, like honest politicians, primes are rare; the number of primes π(n) less than or equal n is o(n). Rosser and Schoenfeld [Rosser et al. 1962, p. 64] wrote that first Legendre (1808) and later Gauss (1849) conjectured that the density of primes should be logarithmic; that is
Rosser and Schoenfeld start with this observation to obtain approximations for sums of the form
The special case f(p) = 1 is the celebrated theorem as follows.
The number π(n) of primes less than or equal to n is asymptotic to 
as n → ∞; that is,
The distance between consecutive primes increases much faster than n as n → ∞ so that the density of primes is 0.
We will state without proof several refinements of the the prime number theorem from [Rosser and Schoenfeld 1962], which we will use in Chapter 16.
Lemma 5.2.
Let n denote the set of primes ≤n.
a) If n ≥ ...
