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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 16 (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 generalization of the BarbanDavenportHalberstam Theorem to number fields
 J. Number Theory
"... Abstract. For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all q ≤ Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the ..."
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Cited by 4 (4 self)
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Abstract. For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all q ≤ Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the BarbanDavenportHalberstam Theorem. 1.
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 2 (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)