Zobrazeno 1 - 10
of 23
pro vyhledávání: '"David J. Platt"'
Publikováno v:
Journal of Number Theory. 238:740-762
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:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::02b5bf2646e37125812389d7f6e27264
https://doi.org/10.1016/j.jnt.2021.09.016
https://doi.org/10.1016/j.jnt.2021.09.016
Publikováno v:
Mathematics of Computation. 90:2923-2935
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:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::2cf92944d99548315080ce58be18d584
https://doi.org/10.1090/mcom/3652
https://doi.org/10.1090/mcom/3652
Autor:
David J. Platt, Tim Trudgian
Publikováno v:
Bulletin of the London Mathematical Society. 53:792-797
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_________::8aa75ca4ffc3a7814decc3842b7ade7b
https://doi.org/10.1112/blms.12460
https://doi.org/10.1112/blms.12460
Autor:
Tim Trudgian, David J. Platt
Publikováno v:
Mathematics of Computation. 90:871-881
We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a long-stan
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_________::785727b03c1f73818a77ad46611ba4cf
https://doi.org/10.1090/mcom/3583
https://doi.org/10.1090/mcom/3583
Autor:
Tim Trudgian, David J. Platt
Publikováno v:
Platt, D J & Trudgian, T 2019, ' Fujii's development on Chebyshev's conjecture ', International Journal of Number Theory, vol. 15, no. 3, pp. 639-644 . https://doi.org/10.1142/S1793042119500337
Chebyshev presented a conjecture after observing the apparent bias towards primes congruent to [Formula: see text]. His conjecture is equivalent to a version of the Generalized Riemann Hypothesis. Fujii strengthened this conjecture; we strengthen it
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::bbfa36ba7b4aa581bad2048ecc35b9dc
https://doi.org/10.1142/s1793042119500337
https://doi.org/10.1142/s1793042119500337
Autor:
Tim Trudgian, David J. Platt
Publikováno v:
Springer Proceedings in Mathematics & Statistics ISBN: 9783030365677
Brun’s constant is \(B=\sum _{p \in P_{2}} p^{-1} + (p+2)^{-1}\), where the summation is over all twin primes. We improve the unconditional bounds on Brun’s constant to \(1.840503< B < 2.288490\), which are about 13% tighter.
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_________::d8e84ac98dfca858507ef0195b98f33f
https://doi.org/10.1007/978-3-030-36568-4_25
https://doi.org/10.1007/978-3-030-36568-4_25
Autor:
David J. Platt, Andrew R. Booker
Publikováno v:
Booker, A & Platt, D J 2019, ' Turing’s Method for the Selberg Zeta-Function ', Communications in Mathematical Physics, vol. 365, no. 1, pp. 295-328 . https://doi.org/10.1007/s00220-018-3243-4
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:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::d8303761ba2a2ea66d7d2763cdf6d4ce
https://doi.org/10.1007/s00220-018-3243-4
https://doi.org/10.1007/s00220-018-3243-4
Autor:
David J. Platt
Publikováno v:
Platt, D J 2017, ' Isolating some non-trivial zeros of Zeta ', Mathematics of Computation, vol. 86, no. 307, pp. 2449-2467 . https://doi.org/10.1090/mcom/3198
We describe a rigorous algorithm to compute Riemann's zeta function on the half line and its use to isolate the non-trivial zeros of zeta with imaginary part ≤ 30,610,046,000 to an absolute precision of ±2-102. In the process, we provide an indepe
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::dbaca079ad2193bc3f23fe281fcbc130
https://doi.org/10.1090/mcom/3198
https://doi.org/10.1090/mcom/3198
Autor:
David J. Platt, Olivier Ramaré
Publikováno v:
Platt, D J & Ramaré, O 2017, ' Explicit estimates : from Λ( n ) in arithmetic progressions to Λ( n) / n ', Experimental Mathematics, vol. 26, no. 1, pp. 77-92 . https://doi.org/10.1080/10586458.2015.1123124
We denote by ψ(x; q, a) the sum of Λ(n)/n for all n≤x and congruent to a mod q and similarly by ψ(x; q, a) the sum of Λ(n) over the same set. We show that the error term in ψ(x; q, a) − (log x)/ϕ(q) − C(q, a) for a suitable constant C(q,
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::9614a1bd79857033f7c61a291df32bd7
https://doi.org/10.1080/10586458.2015.1123124
https://doi.org/10.1080/10586458.2015.1123124
Autor:
Andrew R. Booker, Jonathan Bober, Brian Conrey, Thomas Oliver, Sandro Bettin, Giuseppe Molteni, Raphael S. Steiner, David J. Platt, Min Lee
Publikováno v:
Bettin, S, Bober, J, Booker, A, Conrey, B, Lee, M, Molteni, G, Oliver, T, Platt, D J & Steiner, R 2018, ' A conjectural extension of Hecke’s converse theorem ', Ramanujan Journal, vol. 47, no. 3, pp. 659-684 . https://doi.org/10.1007/s11139-017-9953-y
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:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::9552017a0d66e8527e033a936ffe6421
http://hdl.handle.net/11567/896404
http://hdl.handle.net/11567/896404