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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
Variance of distribution of primes in residue classes
 Quart. J. Math. Oxford Ser
, 1996
"... THE prime number theorem for arithmetic progressions tells us that, for integers a, q s = 1 with (a, q) = 1 we have ..."
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Cited by 6 (3 self)
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THE prime number theorem for arithmetic progressions tells us that, for integers a, q s = 1 with (a, q) = 1 we have
SOME REMARKS ON ALMOST PERIODIC TRANSFORMATIONS
"... proved a number of interesting theorems on “recurrent ” and “almost periodic ” homeomorphisms of a space on itself, In the first part of the present note we give very simple proofs of some of Got&chalk’s theorems in an even more general form. In the second half we consider “regular ” transformations ..."
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proved a number of interesting theorems on “recurrent ” and “almost periodic ” homeomorphisms of a space on itself, In the first part of the present note we give very simple proofs of some of Got&chalk’s theorems in an even more general form. In the second half we consider “regular ” transformations in more detail. 1. Recurrent and almost periodic transformations. Notations. Let f be a continuous mapping (not necessarily a homeomorphism) of a topological space X in itself (that is,f(X) CX). We say that f is recurrent at a point xEX, or that z is recurrent under f, if, given any neighbourhood U(x) of x, there exist infinitely many positive integers n for which ME U(x). (This definition is equivalent to Gottschalk’s if X is a T1 space.) Further, f is almost periodic at x if, given any U(x), there exists an N(x, U(x))> 0 such that for the (infinite) sequence fn;] of positive integers for which f”‘(x) f U(X) we have %;+I ni 5 N.
EVASIVENESS AND THE DISTRIBUTION OF PRIME NUMBERS
, 2010
"... Abstract. A Boolean function on N variables is called evasive if its decisiontree complexity is N. A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n. We confirm the eventual evasiveness of several classes of monotone graph properties under widely ..."
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Abstract. A Boolean function on N variables is called evasive if its decisiontree complexity is N. A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n. We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla’s conjecture on Dirichlet primes implies that (a) for any graph H, “forbidden subgraph H” is eventually evasive and (b) all nontrivial monotone properties of graphs with ≤ n 3/2−ǫ edges are eventually evasive. (n is the number of vertices.) While Chowla’s conjecture is not known to follow from the Extended Riemann Hypothesis (ERH, the Riemann Hypothesis for Dirichlet’s L functions), we show (b) with the bound O(n 5/4−ǫ) under ERH. We also prove unconditional results: (a ′ ) for any graph H, the query complexity of “forbidden subgraph H ” is ` ´ n −O(1); (b) for some constant c> 0, all nontrivial monotone