Distribution of Binomial Coefficients and Digital Functions

Abstract

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.

Divisibility of binomial coefficients by powers of two

“…However we noted in [9] that the trivariate function considered there turns out to be a limit case which has to be treated separately. As this function possesses, up to a factor, the same denominator as our function M (2) , the same restriction applies here.) Set G(x, y, z) := 4yz + xz 2 2y + 2xy 2 z 1−4yz 2xy(1 + yz) .…”

“…. , p − 1, containing at least one symbol = 0, and a nonnegative integer n, we define |n| w as the number of times the word w occurs as a contiguous subword of the binary expansion of n. It was proved by Rowland [26], and implicitely by Barat and Grabner [2], that ϑ p (j, n)/ϑ p (0, n) is given by a polynomial P j in the variables X w , where w are certain finite words in {0, . .…”

“…The above formulas are exact ; in this paper we want to consider matters from a more analytical point of view and we are interested in asymptotic properties of divisibility of binomial coefficients. For some asymptotic results on binomial coefficients modulo primes and prime powers we refer the reader to the papers by Holte [19], Barat and Grabner [2,3] and Greinecker [18]. Our main result is related to a theorem proved by Singmaster [27] saying that any given integer d divides almost all binomial coefficients.…”

An explicit generating function arising in counting binomial coefficients divisible by powers of primes

“…Rowland [32] used an approach very similar to Barat and Grabner's [5] (see also Spearman and Williams [37]) in order to obtain Theorem 0. More precisely, it follows from the proof of this theorem that the monomials X w (1) · · · X w (ℓ) occurring in the polynomial P j satisfy…”

“…The following statement is an easy reformulation of [32,Theorem 2]. The method used for proving this theorem is very similar to the method used in the older paper [5,Theorem 5], which proves a less detailed form of the result, but can be adapted to yield the full statement. See also Remark 1.…”