Related papers: Some examples of well-behaved Beurling number syst…

In this paper we present the experimental results that more clearly than any theory suggest an answer to the question: when in detection of large (probably) prime numbers to apply, a very resource demanding, Miller-Rabin algorithm. Or, to…

Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(\Delta,x_0)$ such that for all $x \geq x_0$ there exists at least one prime in the interval $(x(1 - \Delta^{-1}), x]$.

We find asymptotics for $S_{K,c}(x)$, the number of positive integers below $x$ whose number of prime factors is $c \; \mathrm{mod}\; K$. We study this question in the context of Beurling integers.

We describe a new algorithm for computing the ideal class group, the regulator and a system of fundamental units in number fields under the generalized Riemann hypothesis. We use sieving techniques adapted from the number field sieve…

We show the zero-density estimate \[ N(\zeta_{\mathcal{P}}; \alpha, T) \ll T^{\frac{4(1-\alpha)}{3-2\alpha-\theta}}(\log T)^{9} \] for Beurling zeta functions $\zeta_{\mathcal{P}}$ attached to Beurling generalized number systems with…

We study the distribution of zeros of zeta functions associated to Beurling generalized prime number systems whose integers are distributed as $N(x) = Ax + O(x^{\theta})$. We obtain in particular \[ N(\alpha, T) \ll…

We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{\alpha,\beta}$, where $\alpha>1$ is irrational and of finite type.

By considering the prime zeta function, the author intended to demonstrate in that the Riemann zeta function zeta(s) does not vanish for Re(s)>1/2, which would have proven the Riemann hypothesis. However, he later realised that the proof of…

We consider generalized Stirling numbers of the second kind $% S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right) $, $% k=0,1,\ldots .rp+\sum_{s=2}^{L}r_{s}p_{s}$, where $a,b,\alpha_{s},\beta_{s} $ are complex numbers, and…

Let $ \lfloor {x} \rfloor $ denote the greatest integer less than or equal to a real number $x$. Given real numbers $0<\alpha_1 < \alpha_2 < \cdots< \alpha_k < 1$ satisfying a certain condition, we show that there are infinitely many…

The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of…

We show that there exists a one-to-one correspondence between the set of number-conserving cellular automata (CA) with $q$ inputs and the set of balanced sequences with $q$ terms. This allows to enumerate number-conserving CA. We also show…

We consider a one dimensional affine switched system obtained from a formal limit of a two dimensional linear system. We show this is equivalent to minimising the average digit in beta representations with unrestricted digits. We give a…

We present maximally-fast numerical algorithms for conserved coarsening systems that are stable and accurate with a growing natural time-step $\Delta t=A t_s^{2/3}$. For non-conserved systems, only effectively finite timesteps are…

Beurling slow variation is generalized to Beurling regular variation. A Uniform Convergence Theorem, not previously known, is proved for those functions of this class that are measurable or have the Baire property. This permits their…

Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}.

We consider the classical $k$-means clustering problem in the setting bi-criteria approximation, in which an algoithm is allowed to output $\beta k > k$ clusters, and must produce a clustering with cost at most $\alpha$ times the to the…

We provide an effective upper bound for positive integers with bounded Hamming weights with respect to both a linear recurrence numeration system and an Ostrowski-$\alpha$ numeration system, where $\alpha$ is a quadratic irrational. We…

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…

We prove Beurling's theorem for the full group $SL(2,\R)$. This is the {\em master theorem} in the quantitative uncertainty principle as all the other theorems of this genre follow from it.