DORIN GHISA Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory,algebra, complex analysis, statistics, as well as in physics. Another reason why this function has drawn so much attention is the celebrated Riemann conjecture regarding its non trivial zeros, which resisted proof or disproof until now. There has been lately a lot of progress in the sense of proving its truthfulness.

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The Riemann zeta function

In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

A. Dedekind Sums

Multiplicative Number Theory I: Zeros

The Hardy–Littlewood method: Bibliography

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.

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

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.

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A Survey of Gcd-Sum Functions

We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd convolution. We define and study the convolutes of arithmetic functions of several variables, according to the different types of convolutions. We discuss the multiple Dirichlet series and Bell series and present certain arithmetic and asymptotic results of some special multiplicative functions arising from problems in number theory, group theory, and combinatorics. We give a new proof to obtain the asymptotic density of the set of ordered r-tuples of positive integers with pairwise relatively prime components and consider a similar question related to unitary divisors.

Multiplicative Arithmetic Functions of Several Variables: A Survey

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.

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The discrete Fourier transform of $r$-even functions

Let χ be a Dirichlet character (mod n) with conductor d. In a quite recent paper Zhao and Cao deduced the identity ∑k=1n(k − 1,n)χ(k) = φ(n)τ(n/d), which reduces to Menon’s identity if χ is the principal character (mod n). We generalize the above identity by considering even functions (mod n), and offer an alternative approach to prove. We also obtain certain related formulas concerning Ramanujan sums.

László Tóth

Papers

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

A Survey of Gcd-Sum Functions

Multiplicative Arithmetic Functions of Several Variables: A Survey

The discrete Fourier transform of $r$-even functions

Menon-type identities concerning Dirichlet characters