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Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K su ..."
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Cited by 17 (1 self)
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Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree1 prime p of K such that p = σ, satis
A BombieriVinogradov theorem for all number fields. submitted for publication
"... The classical theorem of Bombieri and Vinogradov is generalized to a nonabelian, nonGalois setting. This leads to a prime number theorem of “mixedtype ” for arithmetic progressions “twisted ” by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murt ..."
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Cited by 1 (1 self)
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The classical theorem of Bombieri and Vinogradov is generalized to a nonabelian, nonGalois setting. This leads to a prime number theorem of “mixedtype ” for arithmetic progressions “twisted ” by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the BombieriVinogradov theorem. We develop this theory with a view to applications in the study of the Euclidean algorithm in number fields and arithmetic orbifolds. Dirichlet’s density theorem gives an asymptotic estimate for the density of primes in arithmetic progressions. Let π(x) denote the number of primes p ≤ x, and for positive integers a ≤ q such that (a, q) = 1, denote by π(x, q, a) the number of primes p ≤ x which satisfy the congruence p ≡ a (mod q). Dirichlet’s theorem indicates that as x → ∞ π(x, q, a) ∼ π(x) φ(q) where φ is Euler’s totient function, and f ∼ g means that f/g → 1. The Riemann hypothesis for all Dirichlet Lfunctions implies that the error term satisfies the estimate π(x) 1 ∣π(x, q, a) − φ(q) ∣ ≪ x 2 log qx, where f ≪ g (equivalently f = O(g)) means that f/g  is bounded, and will be referred to by saying that f is of order g. The celebrated theorem of Bombieri [3] and Vinogradov [24] shows that this estimate holds on the average. Theorem 0.1 (Bombieri, Vinogradov). Let A> 0 be given. Then there is a B = B(A)> 0 so that for Q = x 1 2 (log x) −B max (a,q)=1 max ∣π(y, q, a) − y≤x π(y) φ(q)
A BOMBIERIVINOGRADOV THEOREM
"... Abstract. The classical theorem of Bombieri and Vinogradov is generalized to a nonabelian, nonGalois setting. This leads to a prime number theorem of “mixedtype ” for arithmetic progressions “twisted ” by splitting conditions in number fields. One can view this as an extension of earlier work of ..."
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Abstract. The classical theorem of Bombieri and Vinogradov is generalized to a nonabelian, nonGalois setting. This leads to a prime number theorem of “mixedtype ” for arithmetic progressions “twisted ” by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the BombieriVinogradov theorem. We develop this theory with a view to applications in the study of the Euclidean algorithm in number fields and arithmetic orbifolds. Dirichlet’s density theorem gives an asymptotic estimate for the density of primes in arithmetic progressions. Let π(x) denote the number of primes p ≤ x, andfor positive integers a ≤ q such that (a, q) = 1, denote by π(x, q, a) thenumberof primes p ≤ x which satisfy the congruence p ≡ a (mod q). Dirichlet’s theorem indicates that as x →∞, π(x, q, a) ∼ π(x) φ(q), where φ is Euler’s totient function, and f ∼ g means that f/g → 1. The Riemann hypothesis for all Dirichlet Lfunctions implies that the error term satisfies the estimate π(x, q, a) − π(x)
ℓadic Representations and the Čebotarev Density Theorem
, 2004
"... The Čebotarev Density Theorem, generalizing Dirichlet’s theorem on primes in arithmetic progression, gives us a notion of the density of prime ideals in a number field. We motivate our exposition of the theorem by giving a brief introduction to ℓadic representations, as can be found in [Hus04]. We ..."
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The Čebotarev Density Theorem, generalizing Dirichlet’s theorem on primes in arithmetic progression, gives us a notion of the density of prime ideals in a number field. We motivate our exposition of the theorem by giving a brief introduction to ℓadic representations, as can be found in [Hus04]. We give a brief introduction to class field theory, drawing from [Cox89], and present Deuring’s [Deu35] simple proof of the Čebotarev Density Theorem. We conclude by following the impact of the Čebotarev Density Theorem on ℓadic representations, by looking at ℓadic representations attached to elliptic curves, one of the more famous instances being in the proof of Fermat’s Last Theorem. 1