Sums of weighted averages of gcd-sum functions, II
نویسندگان
چکیده
منابع مشابه
Weighted Gcd-Sum Functions
We investigate weighted gcd-sum functions, including the alternating gcd-sum function and those having as weights the binomial coefficients and values of the Gamma function. We also consider the alternating lcm-sum function.
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We survey properties of the gcd-sum function and of its analogs. As new results, we establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the gcd-sum function and for the function defined by the harmonic mean of the gcd’s.
متن کاملMean Values of Generalized gcd-sum and lcm-sum Functions
We consider a generalization of the gcd-sum function, and obtain its average order with a quasi-optimal error term. We also study the reciprocals of the gcd-sum and lcm-sum functions.
متن کاملMetric discrepancy theory, functions of bounded variation and GCD sums
Let f(x) be a 1-periodic function of bounded variation having mean zero, and let (nk)k≥1 be an increasing sequence of positive integers. Then a result of Baker implies the upper bound ∣∣∣∑Nk=1 f(nkx)∣∣∣ = O (√N(logN)3/2+ε) for almost all x ∈ (0, 1) in the sense of the Lebesgue measure. We show that the asymptotic order of ∣∣∣∑Nk=1 f(nkx)∣∣∣ is closely connected with certain number-theoretic pro...
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(nknl) are established, where (nk)1≤k≤N is any sequence of distinct positive integers and 0 < α ≤ 1; the estimate for α = 1/2 solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for α = 1/2. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues ...
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ژورنال
عنوان ژورنال: Rocky Mountain Journal of Mathematics
سال: 2020
ISSN: 0035-7596
DOI: 10.1216/rmj.2020.50.1045