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

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence a1x1 + · · ·+ akxk ≡ b (mod n), with gcd(xi, n) = ti (1 ≤ i ≤ k), where a1, t1, . . . , ak, tk, b, n (n ≥ 1) are arbitrary integers. As a consequence, we derive necessary and sufficient conditions under which the above restricted linear congruence has no solutions. The number of solutions of these kinds of congruences was first considered by Rademacher in 1925 and Brauer in 1926, in the special case of ai = ti = 1 (1 ≤ i ≤ k). Since then, this problem has been studied, in several other special cases, in many papers. The problem is very well-motivated and has found intriguing applications in several areas of mathematics, computer science, and physics, and there is promise for more applications/implications in these or other directions.

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Restricted linear congruences.

Mills showed that there exists a constant $A$ such that $\lfloor{A^{3^n}}\rfloor$ is prime for every positive integer $n$. Kuipers and Ansari generalized this result to $\lfloor{A^{c^n}}\rfloor$ where $c\in\mathbb{R}$ and $c\geq 2.106$. The main contribution of this paper is a proof that the function $\lceil{B^{c^n}}\rceil$ is also a prime-representing function, where $\lceil X\rceil$ denotes the ceiling or least integer function. Moreover, the first 10 primes in the sequence generated in the case $c=3$ are calculated. Lastly, the value of $B$ is approximated to the first $5500$ digits and is shown to begin with $1.2405547052\ldots$.

A Variation on Mills-Like Prime-Representing Functions

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 Möbius μ-function

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) 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 $k$-compositions of $n$ satisfying certain coprimality conditions

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials. We formulate some basic properties of unitary cyclotomic polynomials and study how they are connected with cyclotomic, inclusion-exclusion and Kronecker polynomials. Further, we derive some related arithmetic function identities involving the unitary analog of the Dirichlet convolution.

László Tóth

Papers

Restricted linear congruences.

A Variation on Mills-Like Prime-Representing Functions

A modified Möbius μ-function

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

Unitary cyclotomic polynomials