Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

Optimality of Chebyshev bounds for Beurling generalized numbers

100%

Diamond H., Zhang W.-B.

Acta Arithmetica

2013

tom 160

nr 3

259-275

If the counting function N(x) of integers of a Beurling generalized number system satisfies both $∫_1^∞ x^{-2}|N(x)-Ax| dx < ∞ $ and $x^{-1}(log x)(N(x)-Ax) = O(1)$, then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that $∫_1^∞ |N(x)-Ax|x^{-2} dx < ∞$ and $x^{-1}(log x)(N(x) - Ax) = O(f(x))$ do not imply the Chebyshev bound.

Chebyshev bounds for Beurling numbers

100%

Diamond H., Zhang W.-B.

Acta Arithmetica

2013

tom 160

nr 2

143-157

The first author conjectured that Chebyshev-type prime bounds hold for Beurling generalized numbers provided that the counting function N(x) of the generalized integers satisfies the L¹ condition $∫_1^∞ |N(x) - Ax| dx/x^2 < ∞$ for some positive constant A. This conjecture was shown false by an example of Kahane. Here we establish the Chebyshev bounds using the L¹ hypothesis and a second integral condition.

Ograniczanie wyników

2 Acta Arithmetica

2 Diamond H.

2 Zhang W.-B.

2 2013

Wyniki wyszukiwania

Optimality of Chebyshev bounds for Beurling generalized numbers

Chebyshev bounds for Beurling numbers