Showing papers by "László Tóth published in 2011"

Menon's identity and arithmetical sums representing functions of several variables

Abstract: We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary orders. We also point out extensions of Menon's identity in the one variable case, which seems to not appear in the literature.

The discrete Fourier transform of $r$-even functions

Abstract: We give a detailed study of the discrete Fourier transform (DFT) of r-even arithmetic functions, which form a subspace of the space of r-periodic arithmetic functions. We consider the DFT of sequences of r-even functions, their mean values and Dirichlet series. Our results generalize properties of the Ramanujan sum. We show that some known properties of r-even functions and of the Ramanujan sum can be obtained in a simple manner via the DFT.

A survey of the alternating sum-of-divisors function

Abstract: We survey arithmetic and asymptotic properties of the alternating sum-of-divisors function $\beta$ defined by $\beta(p^a)=p^a-p^{a-1}+p^{a-2}-...+(-1)^a$ for every prime power $p^a$ ($a\ge 1$), and extended by multiplicativity. Certain open problems are also stated.

Sums of products of Ramanujan sums

Abstract: The Ramanujan sum $c_n(k)$ is defined as the sum of $k$-th powers of the primitive $n$-th roots of unity. We investigate arithmetic functions of $r$ variables defined as certain sums of the products $c_{m_1}(g_1(k))...c_{m_r}(g_r(k))$, where $g_1,..., g_r$ are polynomials with integer coefficients. A modified orthogonality relation of the Ramanujan sums is also derived.

Some remarks on a paper of V. A. Liskovets

Abstract: We deduce new properties of the orbicyclic function $E$ of several variables investigated in a recent paper by V. A. Liskovets. We point out that the function $E$ and its connection to the number of solutions of certain linear congruences occur in the literature in a slightly different form. We investigate another similar function considered by Deitmar, Koyama and Kurokawa by studying analytic properties of some zeta functions of Igusa type. Simple number theoretic proofs for some known properties are also given.

A modified Möbius μ-function

Abstract: We investigate a modified Mobius μ-function which is related to an infinite product of shifted Riemann zeta-functions. We prove conditional and unconditional upper and lower bounds for its summatory function, and, finally, we discuss relations with Riemann’s hypothesis.

A modified M\"obius $\mu$-function

Abstract: We investigate a modified M\"obius $\mu$-function which is related to an infinite product of shifted Riemann zeta-functions. We prove conditional and unconditional upper and lower bounds for its summatory function, and, finally, we discuss relations with Riemann's hypothesis.