76 research outputs found
Goldbach partitions and norms of cusp forms
An integral formula for the Goldbach partitions requires uniform convergence of a complex exponential sum. The dependence of the coefficients of the series is found to be bounded by that of cusp forms. Norms may be defined for these forms on a fundamental domain of a modular group. The relation with the integral formula is found to be sufficient to establish the consistency of the interchange of the integral and the sum, which must remain valid as the even integer tends to infinity
Combinatorial and Additive Number Theory Problem Sessions: '09--'19
These notes are a summary of the problem session discussions at various CANT
(Combinatorial and Additive Number Theory Conferences). Currently they include
all years from 2009 through 2019 (inclusive); the goal is to supplement this
file each year. These additions will include the problem session notes from
that year, and occasionally discussions on progress on previous problems. If
you are interested in pursuing any of these problems and want additional
information as to progress, please email the author. See
http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019,
fixed a few issues from some presenters 6/29/201
Explicit Interval Estimates for Prime Numbers
Using a smoothing function and recent knowledge on the zeros of the Riemann
zeta-function, we compute pairs of such that for all there exists at least one prime in the interval .Comment: 15 pages, 3 tables, 1 figur
Some discussions on the Goldbach conjecture
According to some discussions based on syllogism, we present results on the
binary Goldbach conjecture in three categories: results that are weaker than
the Goldbach conjecture, sufficient conditions for the Goldbach conjecture, and
results that are similar in nature to the Goldbach conjecture. Additionally, we
explore the connections between the Goldbach conjecture and other well-known
conjectures.Comment: Some typos corrected. Many references added. We apologize for any
unintentional omissions and recognize the significant contributions made by
numerous researchers to the study of the Goldbach conjecture. Any comments or
suggestions are welcome
Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences
We consider infinite sequences of superstable orbits (cascades) generated by
systematic substitutions of letters in the symbolic dynamics of one-dimensional
nonlinear systems in the logistic map universality class. We identify the
conditions under which the topological entropy of successive words converges as
a double exponential onto the accumulation point, and find the convergence
rates analytically for selected cascades. Numerical tests of the convergence of
the control parameter reveal a tendency to quantitatively universal
double-exponential convergence. Taking a specific physical example, we consider
cascades of stable orbits described by symbolic sequences with the symmetries
of quasilattices. We show that all quasilattices can be realised as stable
trajectories in nonlinear dynamical systems, extending previous results in
which two were identified.Comment: This version: updated figures and added discussion of generalised
time quasilattices. 17 pages, 4 figure
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