Showing papers by "László Tóth published in 2010"
01 Jan 2010
TL;DR: In this article, the authors survey properties of the gcd-sum function and its analogs and establish asymptotic formulae with remainder terms for the quadratic moment and the reciprocal of the GCS function.
Abstract: 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.
58 citations
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TL;DR: In this paper, the authors investigated the polynomials of the Ramanujan sums and showed connections and analogies to the cyclotomic polynomial polynoms.
Abstract: We investigate the polynomials $\sum_{k=0}^{n-1} c_n(k)x^k$ and $\sum_{k=0}^{n-1} |c_n(k)| x^k$, where $c_n(k)$ denote the Ramanujan sums. We point out connections and analogies to the cyclotomic polynomials.
15 citations
TL;DR: A fourth formula concerning the Euler-type function Φ k is generalized and certain related divisor-type, sum-of-divisors-type and gcd-sum-type functions are investigated.
Abstract: Abstract We give common generalizations of three formulae involving the number of relatively prime subsets of {1, 2, . . . , n} with some additional constraints. We also generalize a fourth formula concerning the Euler-type function Φ k , and investigate certain related divisor-type, sum-of-divisors-type and gcd-sum-type functions.
8 citations
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TL;DR: In this article, an analogue of Ramanujan's sum with respect to regular integers (mod $r$) is introduced and shown to possess properties similar to those of the usual Ramesh's sum.
Abstract: An integer $a$ is said to be regular (mod $r$) if there exists an integer $x$ such that $a^2x\equiv a\pmod{r}$. In this paper we introduce an analogue of Ramanujan's sum with respect to regular integers (mod $r$) and show that this analogue possesses properties similar to those of the usual Ramanujan's sum.
2 citations