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pro vyhledávání: '"Debruyne, Gregory"'
Autor:
Debruyne, Gregory
In this expository article we provide an elegant proof of the one-sided Ingham-Karamata Tauberian theorem. As an application, we present a short deduction of the prime number theorem.
Comment: 7 pages
Comment: 7 pages
Externí odkaz:
http://arxiv.org/abs/2403.17655
We study the range of validity of the density hypothesis for the zeros of $L$-functions associated with cusp Hecke eigenforms $f$ of even integral weight and prove that $N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon}$ holds for $\sigma \geq 1407/16
Externí odkaz:
http://arxiv.org/abs/2310.14797
We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search
Externí odkaz:
http://arxiv.org/abs/2309.01567
Autor:
Broucke, Frederik, Debruyne, Gregory
Publikováno v:
Acta Arith. 207 (2023), 365-391
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 T^{\frac{c(1-\alpha)}{1-\theta}}\l
Externí odkaz:
http://arxiv.org/abs/2211.08716
Publikováno v:
J. Inst. Math. Jussieu 23 (2024), 249-278
We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given $\alpha\in (0,1]$ and $c>0$ (with $c\leq 1$ if $\alpha=1$), a generalized number system is constructe
Externí odkaz:
http://arxiv.org/abs/2109.08509
Publikováno v:
J. Math. Anal. Appl. 494 (2021), Article number 124450
We study the family of Fourier-Laplace transforms $$ F_{\alpha,\beta}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^{\beta}\exp(\mathrm{i} t^{\alpha}-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $\alpha>1$ and $\beta\in\mathbb{C
Externí odkaz:
http://arxiv.org/abs/2005.06116
Publikováno v:
Adv. Math. 370 (2020), Article Numbers 107240
We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies $\pi(x)= \operat
Externí odkaz:
http://arxiv.org/abs/2004.11501
Autor:
Debruyne, Gregory, Tenenbaum, Gérald
Publikováno v:
Indag. Math. (N.S.) 31 (2020), 728-738
We apply the saddle-point method to derive asymptotic estimates or asymptotic series for the number of partitions of a natural integer into parts chosen from a subset of the positive integers whose associated Dirichlet series satisfies certain analyt
Externí odkaz:
http://arxiv.org/abs/2004.05227
Autor:
Debruyne, Gregory
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.
Comment: 5 pages
Comment: 5 pages
Externí odkaz:
http://arxiv.org/abs/2002.00927
Publikováno v:
Proc. Amer. Math. Soc. 149 (2021), 1053-1060
We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a ha
Externí odkaz:
http://arxiv.org/abs/2001.01635