DISTRIBUTION OF BINOMIAL COEFFICIENTS AND DIGITAL FUNCTIONS

2001 ◽  
Vol 64 (3) ◽  
pp. 523-547 ◽  
Author(s):  
GUY BARAT ◽  
PETER J. GRABNER
Keyword(s):  
Arithmetic Functions ◽  
Binomial Coefficients ◽  
Asymptotic Results ◽  
Residue Classes ◽  
Adic Valuation

The distribution of binomial coefficients in residue classes modulo prime powers and with respect to the p-adic valuation is studied. For this purpose, general asymptotic results for arithmetic functions depending on blocks of digits with respect to q-ary expansions are established.

scholarly journals Spatial Equidistribution of Binomial Coefficients Modulo Prime Powers

2016 ◽  
Vol 11 (2) ◽  
pp. 151-161 ◽  
Author(s):  
Guy Barat ◽  
Peter J. Grabner
Keyword(s):  
Spatial Distribution ◽  
Hausdorff Measure ◽  
Empirical Distribution ◽  
Sierpinski Gasket ◽  
Binomial Coefficients ◽  
Sierpiński Gasket ◽  
Residue Classes

AbstractThe spatial distribution of binomial coefficients in residue classes modulo prime powers is studied. It is proved inter alia that empirical distribution of the points (k,m)p−m with 0 ≤ k ≤ n < pm and $\left( {\matrix{n \cr k \cr } } \right) \equiv a\left( {\bmod \;p} \right)^s $ (for (a, p) = 1) for m→∞ tends to the Hausdorff measure on the “p-adic Sierpiński gasket”, a fractals studied earlier by von Haeseler, Peitgen, and Skordev.

scholarly journals Some Asymptotic Results on q-Binomial Coefficients

2016 ◽  
Vol 20 (3) ◽  
pp. 623-634 ◽  
Author(s):  
Richard P. Stanley ◽  
Fabrizio Zanello
Keyword(s):  
Binomial Coefficients ◽  
Asymptotic Results

scholarly journals Higher Moments of Arithmetic Functions in Short Intervals: A Geometric Perspective

2018 ◽  
Vol 2019 (21) ◽  
pp. 6554-6584 ◽  
Author(s):  
Daniel Rayor Hast ◽  
Vlad Matei
Keyword(s):  
Analytic Number Theory ◽  
Arithmetic Functions ◽  
Point Counting ◽  
Fixed Degree ◽  
Asymptotic Results ◽  
Counting Problem ◽  
Short Intervals ◽  
Von Mangoldt Function ◽  
Geometric Explanation ◽  
Intersection Variety

Abstract We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the Möbius function, in short intervals of polynomials over a finite field $\mathbb{F}_{q}$. Using the Grothendieck–Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the ℓ-adic cohomology of these varieties, corresponding to an asymptotic bound on each moment for fixed degree n in the limit as $q \to \infty $. The results of this paper can be viewed as a geometric explanation for asymptotic results that can be proved using analytic number theory over function fields.

SUBSTITUTIONS GENERATING THE FRACTAL MATRICES OF THE p-ADIC VALUATION OF THE BINOMIAL AND LEGENDRE-POLYNOMIAL COEFFICIENTS

Fractals ◽  
2009 ◽  
Vol 17 (04) ◽  
pp. 407-426
Author(s):  
A. BARBÉ ◽  
G. SKORDEV
Keyword(s):  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Block Matrix ◽  
Binomial Coefficients ◽  
Polynomial Coefficients ◽  
Substitution System ◽  
Adic Valuation ◽  
Divisibility Properties

We consider the fractal matrix which describes the divisibility properties of the coefficients of the Legendre polynomials by prime powers, and derive a block matrix substitution system that generates it. Central in the development is a similar substitution for a particular matrix of binomial coefficients.

scholarly journals The p-adic valuation of k-central binomial coefficients

2009 ◽  
Vol 140 (1) ◽  
pp. 31-42 ◽  
Author(s):  
Armin Straub ◽  
Victor H. Moll ◽  
Tewodros Amdeberhan
Keyword(s):  
Binomial Coefficients ◽  
Adic Valuation

Some aspects of familiarization with binomial coefficients

2019 ◽  
pp. 49-53
Author(s):  
Abdulkarim Magomedov ◽  
S.A. Lavrenchenko
Keyword(s):  
Elementary Mathematics ◽  
Binomial Coefficients ◽  
Computational Aspects

New laconic proofs of two classical statements of combinatorics are proposed, computational aspects of binomial coefficients are considered, and examples of their application to problems of elementary mathematics are given.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

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