Zobrazeno 1 - 10
of 53
pro vyhledávání: '"Tim Trudgian"'
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
Autor:
Tim Trudgian, Alessandro Languasco
Publikováno v:
Journal of Number Theory. 236:245-260
Let L ( s , χ ) be the Dirichlet L-function associated to a non-principal primitive Dirichlet character χ defined mod q , where q ≥ 3 . We prove, under the assumption of the Generalised Riemann Hypothesis, the validity of estimates given by Lamzo
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::a31a8f4ef06a5d1ad2e84c757b1b3cf0
https://doi.org/10.1016/j.jnt.2021.07.019
https://doi.org/10.1016/j.jnt.2021.07.019
Publikováno v:
The Ramanujan Journal. 58:913-941
A prime $p$ is a Wolstenholme prime if $\binom{2p}{p}\equiv2$ mod $p^4$, or, equivalently, if $p$ divides the numerator of the Bernoulli number $B_{p-3}$; a Vandiver prime $p$ is one that divides the Euler number $E_{p-3}$. Only two Wolstenholme prim
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::303fe885405a14583a56869f40239dcb
https://doi.org/10.1007/s11139-021-00438-3
https://doi.org/10.1007/s11139-021-00438-3
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:
Tim Trudgian, Michael J. Mossinghoff
Publikováno v:
International Journal of Number Theory. 17:1697-1716
We examine oscillations in a number of sums of arithmetic functions involving [Formula: see text], the total number of prime factors of [Formula: see text], and [Formula: see text], the number of distinct prime factors of [Formula: see text]. In part
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_________::734681ac7f752ecaae1ccb2432984a47
https://doi.org/10.1142/s1793042121500561
https://doi.org/10.1142/s1793042121500561
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:
Mits Kobayashi, Tim Trudgian
Publikováno v:
Journal of Number Theory. 215:138-148
We show that the natural density of positive integers n for which σ ( 2 n + 1 ) ≥ σ ( 2 n ) is between 0.053 and 0.055.
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_________::0b21b38242aac4aa17893f71b29d8d44
https://doi.org/10.1016/j.jnt.2020.02.007
https://doi.org/10.1016/j.jnt.2020.02.007
Publikováno v:
Journal of Number Theory. 215:20-27
We provide an explicit estimate on the least primitive root mod p 2 . We show, in particular, that every prime p has a primitive root mod p 2 that is less than p 0.99 .
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_________::a3e16d177870ba26718e2aafe5bfa3f8
https://doi.org/10.1016/j.jnt.2020.01.002
https://doi.org/10.1016/j.jnt.2020.01.002
Autor:
Tim Trudgian, Michael J. Mossinghoff
Publikováno v:
75 Years of Mathematics of Computation
We consider $\omega(n)$ and $\Omega(n)$, which respectively count the number of distinct and total prime factors of $n$. We survey a number of similarities and differences between these two functions, and study the summatory functions $L(x)=\sum_{n\l
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_________::35696c4c25bc55ef45e1699bfd765a92
https://doi.org/10.1090/conm/754/15149
https://doi.org/10.1090/conm/754/15149