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In this note, we provide an explicit upper bound for $h_K \mathcal{R}_K d_K^{-1/2}$ which depends on an effective constant in the error term of the Ideal Theorem., Comment: 7 pages. Comments are welcome

This paper is a corrigendum to the article ``On the ideal theorem for number fields''. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.

A study of linear Diophantine equations, with a focus on the Frobenius number and the denumerants.

Using estimates on Hooley’s [Formula: see text]-function and a short interval version of the celebrated Dirichlet hyperbola principle, we derive an asymptotic formula for a class of arithmetic functions over short segments. Numerous examples are also given.

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions., Comment: 8 pages

In this note, an upper bound for the sum of fractional parts of certain smooth functions is established. Such sums arise naturally in numerous problems of analytic number theory. The main feature is here an improvement of the main term due to the use of Weyl's bound for exponential sums and a device used by Popov., Comment: 5 pages

Introducing the concept of $$\kappa $$ -factor function generalizing the well-known irrational factor function, we refine a previous estimate given by S. Chaubey, M. Lanius and A. Zaharescu for the irrational factor function in arithmetic progressions. In particular, the dependence in q in the error term is explicited. An asymptotic formula for the $$\kappa $$ -factor function on shifted primes is also given.

We obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} \varphi \left( \left\lfloor{x/n}\right\rfloor\right)$, involving the Euler functions $\varphi$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the reciprocals of integers., Comment: This version incorporate the improvement of the upper bound of Theorem 2.1 in v.1, given by Lixia Dai and Hao Pan in arXiv:1809.10381. Consecutively, Lixia Dai and Hao Pan are now co-authors

Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl's bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials., 6 pages

For a positive integer $n$ let $\mathfrak{P}_n=\prod_{s_p(n)\ge p} p,$ where $p$ runs over all primes and $s_p(n)$ is the sum of the base $p$ digits of $n$. For all $n$ we prove that $\mathfrak{P}_n$ is divisible by all "small" primes with at most one exception. We also show that $\mathfrak{P}_n$ is large, has many prime factors exceeding $\sqrt{n}$, with the largest one exceeding $n^{20/37}$. We establish Kellner's conjecture, which says that the number of prime factors exceeding $\sqrt{n}$ grows asymptotically as $\kappa \sqrt{n}/\log n$ for some constant $\kappa$ with $\kappa=2$. Further, we compare the sizes of $\mathfrak{P}_n$ and $\mathfrak{P}_{n+1}$, leading to the somewhat surprising conclusion that although $\mathfrak{P}_n$ tends to infinity with $n$, the inequality $\mathfrak{P}_n>\mathfrak{P}_{n+1}$ is more frequent than its reverse., Comment: 25 pages

Le present article s'interesse a un celebre probleme de theorie des nombres toujours non resolu: le probleme des diviseurs de Dirichlet. Soit n ≥ 1 un entier et on note τ (n ) le nombre de ses diviseurs positifs. Il est facile de voir que cette fonction τ a un comportement tres erratique. Cet article passe en revue les differentes etudes en moyenne et asymptotique de la fonction τ comme par exemple: ou γ ≈ 0,577 215 664 est la constante d'Euler–Mascheroni, θ ∈ [0,1[ et β ≥ 0 est un reel sans grande importance. Le probleme des diviseurs de Dirichlet consiste alors a determiner le plus petit θ ∈ [0,1[ telle que l'estimation ci-dessus soit vraie.

This chapter is an introduction to algebraic number fields, which arose from both a generalization of the arithmetic in ℤ and the necessity to solve certain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers \(\mathcal {O}_{\mathbb {K}}\) of an algebraic number field \(\mathbb {K}\) is investigated. Next, the ideal numbers, as Kummer called them, are introduced to restore unique factorization. The last section shows how analytic tools can be used to solve hard problems of algebraic number theory. In particular, an account of Zimmert’s method for ideal classes is given.

The theorems of Bachet–Bezout and Gauss are the cornerstones of elementary methods in number theory. Many applications of these results are studied, in particular Diophantine problems. The section Further Developments investigates the number of integer solutions of certain linear Diophantine equations, i.e. the number of certain restricted partitions of an integer.

Some number-theoretic problems involve the number of points with integer coordinates very near some smooth plane curves. This chapter introduces the basic results and some refinements of the theory. Some criteria are investigated and the theorem of Huxley and Sargos is studied in detail. In the section Further Developments, we prove a particular case of a general theorem given by Filaseta and Trifonov improving on the distribution of squarefree numbers in short segments. The dual problem of square-full numbers in short intervals is also investigated.