arXiv subjects

Szilárd Révész

Publications and source records attributed to Szilárd Révész.

4 recordsLinked to original sources

arXiv2024

Some examples of well-behaved Beurling number systems

We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called $[α,β]$-systems, where $α$ and $β$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[α,β]$-system satisfies $\max\{α,β\}\ge1/2$. In this paper we show there are $[α,β]$-systems for each $α\in [0,1)$ and $β\in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[α,β]$-systems with $β<1/2$.

math.NT↗

arXiv2020

Minimal energy point systems on the unit circle and the real line

In this paper, we investigate discrete logarithmic energy problems in the unit circle. We study the equilibrium configuration of $n$ electrons and $n-1$ pairs of external protons of charge $+1/2$. It is shown that all the critical points of the discrete logarithmic energy are global minima, and they are the solutions of certain equations involving Blaschke products. As a nontrivial application, we refine a recent result of Simanek, namely, we prove that any configuration of $n$ electrons in the unit circle is in stable equilibrium (that is, they are not just critical points but are of minimal energy) with respect to an external field generated by $n-1$ pairs of protons.

math.CA↗

arXiv2013

The periodic decomposition problem

If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+α_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation $Δ_{α_1}\dotsΔ_{α_n} f=0$. The periodic decomposition problem asks for the converse implication, which may hold or fail depending on the context (on the system of periods, on the function class in which the problem is considered, etc.). The problem has natural extensions and ramifications in various directions, and is related to several other problems in real analysis, Fourier and functional analysis. We give a survey about the available methods and results, and present a number of intriguing open problems.

math.CA↗

arXiv2013

On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes

We work out the optimization problem, initiated by K. Soundararajan, for the choice of the underlying polynomial P used in the construction of the weight function in the Goldston--Pintz--Yildirim method for finding small gaps between primes. First we reformulate to a maximization problem on L^2[0,1] for a self-adjoint operator T, the norm of which is then the maximal eigenvalue of T. To find eigenfunctions and eigenvalues, we derive a differential equation which can be explicitly solved. The aimed maximal value is S(k)=4/(k+ck^{1/3}), achieved by the (k-1)st integral of x^{1-k/2}J_{k-2}(a_1\sqrt{x}), where a_1 is the first positive root of the (k-2)nd Bessel function J_{k-2} and as such, is asymptotically ck^{1/3} with a well-known constant c. As this naturally gives rise to a number of technical problems in the application of the GPY method, we also construct a polynomial P which is a simpler function yet it furnishes an approximately optimal extremal quantity, 4/(k+Ck^{1/3}) with some other constant C. In the forthcoming paper of J. Pintz (also to appear in the Turán-100 Memorial Volume by de Gruyter), it is indeed shown how this quasi-optimal choice of the polynomial in the weight finally can exploit the GPY method to its theoretical limits.

math.NT↗