Short interval results for certain arithmetic functions [PDF]
, 2016 Using estimates on Hooley’s Δ-function and a short interval version of the celebrated Dirichlet hyperbola principle, we derive an asymptotic formula for a class of arithmetic functions over short segments. Numerous examples are also given.
O. Bordellès
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Metric results on summatory arithmetic functions on Beatty sets [PDF]
Acta Arithmetica, 2019 Let $f\colon\mathbb{N}\rightarrow\mathbb{C}$ be an arithmetic function and consider the Beatty set $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb{N} \,\rbrace$ associated to a real number $\alpha$, where $\lfloor\xi\rfloor ...
Marc Technau, Agamemnon Zafeiropoulos
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MEAN VALUES OF ARITHMETIC FUNCTIONS IN SHORT INTERVALS AND IN ARITHMETIC PROGRESSIONS IN THE LARGE‐DEGREE LIMIT [PDF]
Mathematika, 2018 A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as large as ...
O. Gorodetsky
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Asymptotics of Arithmetic Functions of GCD and LCM of Random Integers in Hyperbolic Regions [PDF]
Results in Mathematics, 2021 We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets
A. Iksanov, A. Marynych, K. Raschel
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On some results concerning generalized arithmetic triangles [PDF]
Electron. Notes Discret. Math., 2018 In this paper we present theoretical and computational results regarding generalized arithmetic m-triangles. The numerical values recover well-known number sequences, indexed in the OEIS including binomial coefficients and their extensions.
A. Bagdasaryan, O. Bagdasar
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Higher moments of arithmetic functions in short intervals: a geometric perspective [PDF]
, 2016 We study the geometry associated to the distribution of certain arithmetic functions, including the von Mangoldt function and the M\"obius function, in short intervals of polynomials over a finite field $\mathbb{F}_q$.
D. Hast, Vlad Matei
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A Sharpening of Effective Formulas of Selberg–Delange Type for Some Arithmetic Functions on the SemigroupGK [PDF]
, 1996 In this paper, we establish effective asymptotic formulas of Delange–Selberg type for some arithmetic functions on the semigroupGKof all non-zero integral ideals in ZK.
Jie Wu
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Asymptotic Formulas for Some Arithmetic Functions [PDF]
Canadian mathematical bulletin, 1958 Let f(x) be an increasing function. Recently there have been several papers which proved that under fairly general conditions on f(x) the density of integers n for which (n, f(n)) = 1 is 6/π2 and that (d(n) denotes the number of divisors of n)In ...
P. Erdős
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Mean values of arithmetic functions on a sparse set and applications [PDF]
Colloquium MathematicumLet f be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish some asymptotic estimates for the quantities ψf(x):=∑n≤xΛ(n)f([xn]),Mπf(x):=∑p≤xf([xp]) for x→∞, where Λ(n) is the von Mangoldt function and [t] is ...
Hengcai Tang, Jie Wu
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On the distribution of the roots of certain congruences and a problem for additive arithmetic functions [PDF]
, 1983 The asymptotic distribution of the roots of the congruence ax ≡ b (mod D), 1 ≤ x ≤ D, as D varies, is investigated. Quantitative estimates are obtained by means of exponential sums combined with sieve methods. As an application of the results it is shown
P. Elliott
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