Chebyshev type estimates in prime number theory.
Interpolation of the Dirichlet divisor problem
On a Derivation Equation in a Measure Algebra.
Characterization of Derivations on an Algebra of Measures.
Characterization of Derivations on an Algebra of Measures.II.
When do Beurling generalized integers have a density?
Metric theorems on the estimation of integrals by sums
Chebyshev bounds for Beurling numbers
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 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.
Optimality of Chebyshev bounds for Beurling generalized numbers
If the counting function N(x) of integers of a Beurling generalized number system satisfies both and , 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 and do not imply the Chebyshev bound.
Oscillation of Mertens’ product formula
Mertens’ product formula asserts that as . Calculation shows that the right side of the formula exceeds the left side for . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on , this and a complementary inequality might change their sense for sufficiently large values of . We show this to be the case.
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