Zobrazeno 1 - 10
of 69
pro vyhledávání: '"Platt, David J."'
Publikováno v:
Mathematics of Computation 90 (2021), 2923-2935
We consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be
Externí odkaz:
http://arxiv.org/abs/2009.13791
Publikováno v:
Bulletin of the Australian Mathematical Society 104 (2021), 59-65
We consider the sum $\sum 1/\gamma$, where $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a smooth ap
Externí odkaz:
http://arxiv.org/abs/2009.05251
We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that $\frac{1}{5\,374}\leq I(X
Externí odkaz:
http://arxiv.org/abs/2008.06140
Publikováno v:
In Journal of Number Theory September 2022 238:740-762
Autor:
Booker, Andrew R., Platt, David J.
In one of his final research papers, Alan Turing introduced a method to certify the completeness of a purported list of zeros of the Riemann zeta-function. In this paper we consider Turing's method in the analogous setting of Selberg zeta-functions,
Externí odkaz:
http://arxiv.org/abs/1710.00603
Autor:
Bettin, Sandro, Bober, Jonathan W., Booker, Andrew R., Conrey, Brian, Lee, Min, Molteni, Giuseppe, Oliver, Thomas, Platt, David J., Steiner, Raphael S.
We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Rama
Externí odkaz:
http://arxiv.org/abs/1704.02570
Autor:
Platt, David J., Trudgian, Timothy S.
Publikováno v:
LMS J. Comput. Math. 19 (2016) 37-41
This article considers the positive integers $N$ for which $\zeta_{N}(s) = \sum_{n=1}^{N} n^{-s}$ has zeroes in the half-plane $\Re(s)>1$. Building on earlier results, we show that there are no zeroes for $1\leq N\leq 18$ and for $N=20, 21, 28$. For
Externí odkaz:
http://arxiv.org/abs/1507.01340
Autor:
Eddin, Sumaia Saad, Platt, David J.
Let $\chi$ be a primitive Dirichlet character of conductor $q$ and $L(z,\chi)$ the associated L-series. In this paper we provide an explicit upper bound for $|L(1, \chi)|$ when 3 divides $q$.
Comment: 11 pages, 2 figures. Submitted to Colloquium
Comment: 11 pages, 2 figures. Submitted to Colloquium
Externí odkaz:
http://arxiv.org/abs/1306.4780
Autor:
Platt, David J.
We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q<=400,000. For eve
Externí odkaz:
http://arxiv.org/abs/1305.3087