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 deduce a simple representation and the invariant factor decompositions of the subgroups of the group $\Bbb{Z}_m \times \Bbb{Z}_n$, where $m$ and $n$ are arbitrary positive integers. We obtain formulas for the total number of subgroups and the number of subgroups of a given order.

On the subgroups of the group Z_m x Z_n

We generalize the asymptotic estimates by Bubboloni, Luca
and Spiga [2] on the number of $$k$$
 -compositions of n satisfying some coprimality
conditions. We substantially refine the error term concerning the number of

 $$k$$
 -compositions of $$n$$
 with pairwise relatively prime summands. We use a different
approach, based on properties of multiplicative arithmetic functions of $$k$$
 variables
and on an asymptotic formula for the restricted partition function.

On the number of $$k$$ k -compositions $$n$$ n satisfying certain coprimality conditions

An integer $k$ is called regular (mod $n$) if there exists an integer $x$ such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$ possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that $k^{m+1} \equiv k$ (mod $n$). Let $\varrho(n)$ denote the number of regular integers (mod $n$) in the set $\{1,2,\ldots,n\}$. This is an analogue of Euler's $\phi$ function. We introduce the multidimensional generalization of $\varrho$, which is the analogue of Jordan's function. We establish identities for the power sums of regular integers (mod $n$) and for some other finite sums and products over regular integers (mod $n$), involving the Bernoulli polynomials, the Gamma function and the cyclotomic polynomials, among others. We also deduce an analogue of Menon's identity and investigate the maximal orders of certain related functions.

Some remarks on regular integers modulo $n$

We study the exact extremal orders of compositions $f(g(n))$ of certain arithmetical functions, including the functions $\sigma(n)$, $\phi(n)$, $\sigma^*(n)$ and $\phi^*(n)$, representing the sum of divisors of $n$, Euler's function and their unitary analogues, respectively Our results complete, generalize and refine known results

Extremal orders of compositions of certain arithmetical functions

We deduce direct formulas for the total number of subgroups and the number of subgroups of a given order of the group $\Bbb{Z}_m\times \Bbb{Z}_n \times \Bbb{Z}_r \times \Bbb{Z}_s$, where $m,n,r,s\in \Bbb{N}$. The proofs are by some simple group theoretical and number theoretical arguments based on Goursat's lemma for groups. Two conjectures are also formulated.

László Tóth

Papers

On the subgroups of the group Z_m x Z_n

On the number of $$k$$ k -compositions $$n$$ n satisfying certain coprimality conditions

Some remarks on regular integers modulo $n$

Extremal orders of compositions of certain arithmetical functions

The number of subgroups of the group $\Bbb{Z}_m\times \Bbb{Z}_n \times \Bbb{Z}_r \times \Bbb{Z}_s$