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 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.

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On certain sums concerning the gcd's and lcm's of $k$ positive integers

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 $${c_{m_1}(g_1(k))\cdots c_{m_r}(g_r(k))}$$
 , 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

Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. As a well known family, one can mention MMH∗ which was shown to be Δ-universal by Halevi and Krawczyk in 1997. In this paper, we first introduce a variant of MMH∗ that we call GRDH. Then via a novel approach, namely, connecting the universal hashing problem to the number of solutions of (restricted) linear congruences, we investigate the e-almost-Δ — universality of GRDH. Finally, as an application of our results, we propose an authentication code with secrecy scheme which generalizes a recent construction.

On a variant of multilinear modular hashing with applications to authentication and secrecy codes

Let $\chi$ be a Dirichlet character (mod $n$) with conductor $d$. In a quite recent paper Zhao and Cao deduced the identity $\sum_{k=1}^n (k-1,n) \chi(k)= \varphi(n)\tau(n/d)$, which reduces to Menon's identity if $\chi$ is the principal character (mod $n$). We generalize the above identity by considering even functions (mod $n$), and offer an alternative approach to proof. We also obtain certain related formulas concerning Ramanujan sums.

Menon-type identities concerning Dirichlet characters

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\Phi_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime factors using numerical semigroups, respectively Bachman's inclusion-exclusion polynomials. Given $m \ge 1$ we show that every integer occurs as a coefficient of $\Phi^*_{mn}(x)$ for some $n\ge 1$ following Ji, Li and Moree [9]. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/\Phi_n^*(x)$, the inverse unitary cyclotomic polynomials.

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László Tóth

Papers

On certain sums concerning the gcd's and lcm's of $k$ positive integers

Sums of products of Ramanujan sums

On a variant of multilinear modular hashing with applications to authentication and secrecy codes

Menon-type identities concerning Dirichlet characters

Coefficients of (inverse) unitary cyclotomic polynomials