Primes in higher-order progressions on average

On primes in arithmetic progressions

Let d > 4 and c ∈ (−d, d) be relatively prime integers, and let r(d) be the product of all distinct prime divisors of d. We show that for any sufficiently large integer n (in particular n > 24310 suffices for 4 6 d 6 36) the least positive integer m with 2r(d)k(dk− c) (k = 1, . . . , n) pairwise distinct modulo m is just the first prime p ≡ c (mod d) with p > (2dn − c)/(d − 1). We also conjectu...

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 and other small moduli.

Primes in arithmetic progressions

[1] Euler’s proof uses only simple properties of ζ(s), and only of ζ(s) as a function of a real, rather than complex, variable. Given the status of complex number and complex analysis in Euler’s time, this is not surprising. It is slightly more surprising that Dirichlet’s original argument also was a real-variable argument, since by that time, a hundred years later, complex analysis was well-es...

Dirichlet’s Theorem on Primes in Arithmetic Progressions

Let us be honest that the proof of Dirichlet’s theorem is of a difficulty beyond that of anything else we have attempted in this course. On the algebraic side, it requires the theory of characters on the finite abelian groups U(N) = (Z/NZ)×. From the perspective of the 21st century mathematics undergraduate with a background in abstract algebra, these are not particularly deep waters. More seri...

1 Arithmetic progressions in primes