Zobrazeno 1 - 10
of 48
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
Publikováno v:
Architecture; Mar89, Vol. 78 Issue 3, p47-47, 1/9p