5 papers · 1 filter
On the connection between zero-free regions and the error term in the Prime Number Theorem
Frederik Broucke
We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and Révész (2024).…
On zero-density estimates for Beurling zeta functions
Frederik Broucke
We show the zero-density estimate \[ N(ζ_{\mathcal{P}}; α, T) \ll T^{\frac{4(1-α)}{3-2α-θ}}(\log T)^{9} \] for Beurling zeta functions attached to Beurling genera…
A Mean Value Theorem for general Dirichlet Series
Frederik Broucke, Titus Hilberdink
In this paper we obtain a mean value theorem for a general Dirichlet series with positive coefficients for which the counting function $A(x)…
Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder
Frederik Broucke
Given , we show the existence of a Beurling generalized number system whose integer counting satisfies for some and $c…
Beurling integers with RH and large oscillation
Frederik Broucke, Gregory Debruyne, Jasson Vindas
We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The pr…