scholarly journals Limit laws for arithmetic functions of J. P. Kubilius class H, defined on the set of „shifted“ primes

1965 ◽  
Vol 5 (1) ◽  
pp. 5-8
Author(s):  
M. B. Barban ◽  
A. I. Vinogradov ◽  
B. V. Levin
Keyword(s):  
Arithmetic Functions ◽  
Limit Laws ◽  
Shifted Primes

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: М. Б. Барбан, А. И. Виноградов, Б. В. Левин. Предельные законы для функций класса Н И. П. Кубилюса, заданных на множестве «сдвинутых» простых чисел M. B. Barbanas, A. I. Vinogradovas, B. V. Levinas. Ribiniai dėsniai J. Kubiliaus klasės H funkcijoms, definuotoms „pastumtų“ pirminių skaičių aibėje

scholarly journals Correlations of Sums of Two Squares and Other Arithmetic Functions in Function Fields

2017 ◽  
Vol 2019 (14) ◽  
pp. 4469-4515 ◽  
Author(s):  
Lior Bary-Soroker ◽  
Arno Fehm
Keyword(s):  
Finite Field ◽  
Function Field ◽  
Function Fields ◽  
Arithmetic Functions ◽  
Quadratic Character ◽  
Field Limit ◽  
Numerical Evidence ◽  
Divisor Functions ◽  
Shifted Primes ◽  
Sums Of Two Squares

Abstract We investigate a function field analogue of a recent conjecture on autocorrelations of sums of two squares by Freiberg, Kurlberg, and Rosenzweig, which generalizes an older conjecture by Connors and Keating. In particular, we provide extensive numerical evidence and prove it in the large finite field limit. Our method can also handle correlations of other arithmetic functions and we give applications to (function field analogues of) the average of sums of two squares on shifted primes, and to autocorrelations of higher divisor functions twisted by a quadratic character.

Stable limit laws for additive arithmetic functions

1985 ◽  
Vol 37 (4) ◽  
pp. 259-263
Author(s):  
N. M. Timofeev
Keyword(s):  
Arithmetic Functions ◽  
Stable Limit ◽  
Additive Arithmetic ◽  
Limit Laws

scholarly journals Arithmetic functions at consecutive shifted primes

2015 ◽  
Vol 11 (05) ◽  
pp. 1477-1498 ◽  
Author(s):  
Paul Pollack ◽  
Lola Thompson
Keyword(s):  
Natural Number ◽  
Arithmetic Functions ◽  
Infinitely Many Solutions ◽  
Shifted Primes

For each of the functions f ∈ {φ, σ, ω, τ} and every natural number K, we show that there are infinitely many solutions to the inequalities f(pn - 1) < f(pn+1 - 1) < ⋯ < f(pn+K - 1), and similarly for f(pn -1) > f(pn+1 - 1) > ⋯ > f(pn+K -1). We also answer some questions of Sierpiński on the digit sums of consecutive primes. The arguments make essential use of Maynard and Tao's method for producing many primes in intervals of bounded length.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

scholarly journals On the growth and zeros of polynomials attached to arithmetic functions

2021 ◽  
Author(s):  
Bernhard Heim ◽  
Markus Neuhauser
Keyword(s):  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Fourier Coefficients ◽  
Arithmetic Functions ◽  
Zeros Of Polynomials ◽  
Simple Complex ◽  
Hook Length ◽  
Moderate Growth ◽  
Growth Properties ◽  
Sum Of Divisors

AbstractIn this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and $$0<h(n) \le h(n+1)$$ 0 < h ( n ) ≤ h ( n + 1 ) . We put $$P_0^{g,h}(x)=1$$ P 0 g , h ( x ) = 1 and $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ P n g , h ( x ) : = x h ( n ) ∑ k = 1 n g ( k ) P n - k g , h ( x ) . As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind $$\eta $$ η -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

scholarly journals On Sinaĭ Billiards on Flat Surfaces with Horns

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Henk Bruin
Keyword(s):  
Exponential Decay ◽  
Height Function ◽  
Suspension Flow ◽  
Limit Laws ◽  
Suspension Flows ◽  
Flat Surfaces ◽  
Exponential Tails ◽  
Exponential Decay Of Correlations ◽  
Sinai Billiards ◽  
Young Towers

AbstractWe show that certain billiard flows on planar billiard tables with horns can be modeled as suspension flows over Young towers (Ann. Math. 147:585–650, 1998) with exponential tails. This implies exponential decay of correlations for the billiard map. Because the height function of the suspension flow itself is polynomial when the horns are Torricelli-like trumpets, one can derive Limit Laws for the billiard flow, including Stable Limits if the parameter of the Torricelli trumpet is chosen in (1, 2).

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