A refinement of a theorem of Schur on primes in arithmetic progressions III
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...
متن کاملDirichlet’s Theorem about Primes in Arithmetic Progressions
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.
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.
متن کاملPrimes 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...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1969
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-15-2-193-197