149 research outputs found
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Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem
In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that
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Generalised prime systems with periodic integer counting function
We study generalised prime systems (both discrete and continuous) for which the `integer counting
function' N(x) has the property that N(x) ¡ cx is periodic for some c > 0. We show that this is
extremely rare. In particular, we show that the only such system for which N is continuous is the
trivial system with N(x) ¡ cx constant, while if N has finitely many discontinuities per bounded
interval, then N must be the counting function of the g-prime system containing the usual primes
except for finitely many.
Keywords and phrases: Generalised prime systems.
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Asymptotic expansions for Taylor coefficients of the composition of two functions
We give an asymptotic expansion for the Taylor coe±cients of L(P(z)) where L(z)
is analytic in the open unit disc whose Taylor coe±cients vary `smoothly' and P(z) is a
probability generating function. We show how this result applies to a variety of problems,
amongst them obtaining the asymptotics of Bernoulli transforms and weighted renewal
sequences
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Flows of Mellin transforms with periodic integrator
We study Mellin transforms ˆ N(s) = ∫ ∞ 1 − x−sdN(x) for which N(x) − x is periodic with period 1 in order to investigate ‘flows ’ of such functions to Riemann’s ζ(s) and the possibility of proving the Riemann Hypothesis with such an approach. We show that, excepting the trivial case where N(x) = x, the supremum of the real parts of the zeros of any such function is at least 1 2. We investigate a particular flow of such functions { ˆ Nλ}λ≥1 which converges locally uniformly to ζ(s) as λ → 1, and show that they exhibit features similar to ζ(s). For example, ˆ Nλ(s) has roughly T T T log − zeros in the critical strip up to height T and an infinite number of negative zeros
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An optimization problem concerning multiplicative functions
In this paper we study the problem of maximizing a quadratic form 〈Ax,x〉 subject to ‖x‖q=1, where A has matrix entries View the MathML source with i,j|k and q≥1. We investigate when the optimum is achieved at a ‘multiplicative’ point; i.e. where x1xmn=xmxn. This turns out to depend on both f and q, with a marked difference appearing as q varies between 1 and 2. We prove some partial results and conjecture that for f multiplicative such that 0<f(p)<1, the solution is at a multiplicative point for all q≥1
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Quasi Kronecker products and a determinant formula
We introduce an extension of the Kronecker product for matrices which retains many of the properties of the usual Kronecker product. As an application we study matrices over divisor-closed sets with multiplicative entries, and show how these are quasi Kronecker products over the primes of simpler matrices. In particular this gives a formula for the determinant of such matrices which combines and generalizes a number of previous results on Smith type determinants
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Singular values of multiplicative Toeplitz matrices
We study the asymptotic behaviour of the singular values of matrices with entries if and zero otherwise,
with an arithmetical function. In particular, we study the case where is multiplicative and is regularly varying. Our main result is that, under quite general conditions, the singular values are, asymptotically, , where are the eigenvalues of some positive Hilbert-Schmidt operator
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