Lecture # 2: Dirichlet L-functions, Dirichlet Characters and primes in arithmetic progressions
نویسنده
چکیده
Dirichlet considered a question very similar to the one which inspired Euler’s introduction of the ζfunction: namely, how the primes are distributed modulo m. The simplest question of this type is whether are there infinitely many primes congruent to a modulo m. Obviously there can only be infinitely many primes of this form if a and m are relatively prime. Unfortunately, this problem turned out to be much more difficult than proving that there are infinitely many primes. Elementary results along the lines of Euclid’s proof only sufficed to show very special cases. For example, Proposition 1.1. There are infinitely many primes p ≡ 3 (mod 4). Proof. Suppose there were only finitely many such primes, p1, p2, . . . , pn. Consider the number Q = 4p1p2 . . . pn − 1. Clearly this number is not divisible by any of the primes which are 3 modulo 4. Thus, Q ≡ 3 (mod 4) is a product of primes all of which are 1 modulo 4. This is clearly a contradiction.
منابع مشابه
18.785F17 Number Theory I Lecture 18 Notes: Dirichlet L-functions, Primes in Arithmetic Progressions
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18.785F16 Number Theory I Lecture 18 Notes: Dirichlet L-functions, Primes in Arithmetic Progressions
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