Showing papers by "László Tóth published in 2018"
TL;DR: It is proved that the family GRDH is an $\varepsilon$-almost-$\Delta$-universal family of hash functions for some $\vAREpsilon<1$ if and only if $n$ is odd and $\gcd(x_i,n)=t_i=1$ $(1\leq i-leq k)$.
Abstract: Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. MMH∗, which was shown to be Δ-universal by Halevi and Krawczyk in 1997, is a well-kn...
15 citations
9 citations
TL;DR: In this paper, the authors deduce asymptotic formulas for the sums ∑ n 1, r ≤ x f ( n 1 ⋯ n r ) and ∑ r ≥ 2 ≤ x r ≤ r f ( [ n 1, r, n r ] ), where r is a fixed integer, r is the least common multiple of the integers n 1 and r is one of the divisor functions τ 1, k ( n) (k ≥ 1 ), τ ( e ) ( n ) and τ ⁎ ( n )).
7 citations
TL;DR: In this article, the authors generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables, and show that some properties on expansions of one and several variables using classical and unitary sums, respectively, run parallel.
Abstract: We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel.
4 citations
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TL;DR: In this article, it was shown that the ceiling function is also a prime-representing function, and the first 10 primes in the sequence generated in the case $c = 3 were calculated.
Abstract: Mills showed that there exists a constant $A$ such that $\lfloor{A^{3^n}}\rfloor$ is prime for every positive integer $n$. Kuipers and Ansari generalized this result to $\lfloor{A^{c^n}}\rfloor$ where $c\in\mathbb{R}$ and $c\geq 2.106$. The main contribution of this paper is a proof that the function $\lceil{B^{c^n}}\rceil$ is also a prime-representing function, where $\lceil X\rceil$ denotes the ceiling or least integer function. Moreover, the first 10 primes in the sequence generated in the case $c=3$ are calculated. Lastly, the value of $B$ is approximated to the first $5500$ digits and is shown to begin with $1.2405547052\ldots$.
4 citations
TL;DR: In this paper, the authors consider inertia, positive definiteness and l p norm of GCD and LCM matrices and their unitary analogs, and prove that the GCD matrix (S ) f on S associated with f is defined as the n × n matrix having f evaluated at the greatest common divisor of x i and x j as its ij entry.
3 citations
TL;DR: In this article, the authors use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of positive integers, where the lcm is fixed.
Abstract: We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordell\`{e}s (2007), and extend certain related results of Hilberdink and T\'oth (2016). We also formulate some conjectures and open problems.
3 citations
TL;DR: In this paper, a simple proof and a generalization of a Menon-type identity by Li, Hu and Kim, involving Dirichlet characters and additive characters, is presented.
Abstract: We present a simple proof and a generalization of a Menon-type identity by Li, Hu and Kim, involving Dirichlet characters and additive characters.
1 citations
TL;DR: The authors generalize a Menon-type identity by Li and Kim involving additive characters of the group ${\Bbb Z}_n$ using a different approach, based on certain convolutional identities.
Abstract: By considering even functions (mod $n$) we generalize a Menon-type identity by Li and Kim involving additive characters of the group ${\Bbb Z}_n$. We use a different approach, based on certain convolutional identities. Some other applications, including related formulas for Ramanujan sums, are discussed, as well.
TL;DR: In this paper, the additive subgroups of the ring are represented as functions of two variables and their Dirichlet series can be expressed in terms of the Riemann zeta function.
Abstract: Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$ and its unital subrings, respectively. We show that the functions $(m,n)\mapsto N^{(s)}(m,n)$ and $(m,n)\mapsto N^{(us)}(m,n)$ are multiplicative, viewed as functions of two variables, and their Dirichlet series can be expressed in terms of the Riemann zeta function. We also establish an asymptotic formula for the sum $\sum_{m,n\le x} N^{(s)}(m,n)$, the error term of which is closely related to the Dirichlet divisor problem.