Results 1 to 10 of about 5,332 (130)
On a Dynamical Approach to Some Prime Number Sequences. [PDF]
Entropy (Basel), 2018 We show how the cross-disciplinary transfer of techniques from dynamical systems theory to number theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns ...
Lacasa L +3 more
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Some remarks on the first Hardy-Littlewood conjecture [PDF]
Atti della Società dei Naturalisti e Matematici di Modena, 1974 Starting from the first Hardy-Littlewood conjecture some topics will be covered: an empirical approach to the distribution of the twin primes in classes mod(10) and a simplified proof of the Bruns theorem .
Bortolomasi, Marco, Ortiz-Tapia, Arturo
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On certain inequalities for the prime counting function – Part III [PDF]
Notes on Number Theory and Discrete Mathematics, 2023 As a continuation of [10] and [11], we offer some new inequalities for the prime counting function π(x). Particularly, a multiplicative analogue of the Hardy–Littlewood conjecture is provided. Improvements of the converse of Landau's inequality are given.
József Sándor
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On the calculation of integer sequences, associated with twin primes
Lietuvos Matematikos Rinkinys, 2023 The twin primes conjecture states that there are infinitely many twin primes. While studying this hypothesis, many important results were obtained, but the problem remains unsolved.
Igoris Belovas +2 more
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On the Hardy–Littlewood–Chowla conjecture on average
Forum of Mathematics, Sigma, 2022 AbstractThere has been recent interest in a hybrid form of the celebrated conjectures of Hardy–Littlewood and of Chowla. We prove that for any$k,\ell \ge 1$and distinct integers$h_2,\ldots ,h_k,a_1,\ldots ,a_\ell $, we have:$$ \begin{align*}\sum_{n\leq X}\mu(n+h_1)\cdots \mu(n+h_k)\Lambda(n+a_1)\cdots\Lambda(n+a_{\ell})=o(X)\end{align*} $$for all ...
Jared Duker Lichtman, Joni Teräväinen
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The Hardy–Littlewood conjecture and rational points [PDF]
Compositio Mathematica, 2014 AbstractSchinzel’s Hypothesis (H) was used by Colliot-Thélène and Sansuc, and later by Serre, Swinnerton-Dyer and others, to prove that the Brauer–Manin obstruction controls the Hasse principle and weak approximation on pencils of conics and similar varieties. We show that when the ground field is $\mathbb{Q}$ and the degenerate geometric fibres of the
Harpaz, J. +2 more
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A note on the primality of sums [PDF]
Surveys in Mathematics and its Applications, 2022 It is shown that when adding a large number to a set of much smaller numbers, the number of primes or twin ranks (see text) in the resulted sumset can be substantially larger than the theoretical values given by the Prime Number Theorem or Hardy ...
Antonie Dinculescu
doaj
Hidden structure in the randomness of the prime number sequence? [PDF]
, 2005 We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would ...
Bays +17 more
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The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero
Journal of the London Mathematical Society, 2022 Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$, we establish an asymptotic formula for $$\sum_{n\leq x}Λ(n+h_1)\cdots Λ(n+h_k)λ(n+h_{1}')\cdots λ(n+h_{\ell ...
Tao, Terence, Teräväinen, Joni
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Bounded gaps between primes in number fields and function fields [PDF]
, 2014 The Hardy--Littlewood prime $k$-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress ...
Castillo, Abel +4 more
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