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

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.

Sums of products of Ramanujan sums

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.

Some remarks on a paper of V. A. Liskovets

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.

Ramanujan expansions of arithmetic functions of several variables

We give common generalizations of the Menon-type identities by Sivaramakrishnan (1969) and Li, Kim, Qiao (2019). Our general identities involve arithmetic functions of several variables, and also contain, as special cases, identities for gcd-sum type functions. We point out a new Menon-type identity concerning the lcm function. We present a simple character free approach for the proof.

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Menon-type identities again: A note on a paper by Li, Kim and Qiao

We define the k-dimensional generalized Euler function $$\varphi _k(n)$$
 as the number of ordered k-tuples $$(a_1,\ldots ,a_k)\in {\mathbb {N}}^k$$
 such that $$1\le a_1,\ldots ,a_k\le n$$
 and both the product $$a_1\cdots a_k$$
 and the sum $$a_1+\cdots +a_k$$
 are prime to n. We investigate some of the properties of the function $$\varphi _k(n)$$
 , and obtain a corresponding Menon-type identity.

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László Tóth

Papers

Sums of products of Ramanujan sums

Some remarks on a paper of V. A. Liskovets

Ramanujan expansions of arithmetic functions of several variables

Menon-type identities again: A note on a paper by Li, Kim and Qiao

Another generalization of Euler’s arithmetic function and Menon’s identity