A refinement of a theorem of Schur on primes in arithmetic progressions II
نویسندگان
چکیده
منابع مشابه
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...
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Dirichlet’s theorem states that if q and l are two relatively prime positive integers, there are infinitely many primes of the form l+kq. Dirichlet’s theorem is a generalized statement about prime numbers and the theory of Fourier series on the finite abelian group (Z/qZ)∗ plays an important role in the solution.
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The third moment ∑ q≤Q q ∑ a=1 ( ψ(x; q, a)− ρ(x; q, a) )3 is investigated with the novel approximation ρ(x; q, a) = ∑ n≤x n≡a (mod q) FR(n),
متن کاملOn 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...
متن کاملPrimes 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.
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1966
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-12-1-97-109