The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, let P(n) stand for the largest prime factor of n. Then define f(x,p): = #{n ≤ x | P(n) = p}. If f(x,p) is considered as a function of p, for 2 ≤ p ≤ x, the primes in the interval [2,x] belong to three intervals I₁(x) = [2,v(x)], I₂(x) = ]v(x),w(x)[ and I₃(x) = [w(x),x], with v(x) < w(x), such that f(x,p) increases for p ∈ I₁(x), reaches its maximum value in...

We explicitly determine the elliptic $K3$ surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from $2$, for each of the two possible maximal fibre types, $I_{19}$ and $I_{14}^{*}$, the surface is unique. In characteristic $2$ the maximal fibre types are $I_{18}$ and $I_{13}^{*}$, and there exist two (resp. one) one-parameter families of such surfaces.

Lately, explicit upper bounds on $\left|L\right(1,\chi \left)\right|$ (for primitive Dirichlet characters $\chi$) taking into account the behaviors of $\chi$ on a given finite set of primes have been obtained. This yields explicit upper bounds on residues of Dedekind zeta functions of abelian number fields taking into account the behavior of small primes, and it as been explained how such bounds yield improvements on lower bounds of relative class numbers of CM-fields whose maximal totally real subfields are abelian. We present here some other...

This article deals with the value distribution of multiplicative prime-independent arithmetic functions $\left(\alpha \right(n\left)\right)$ with $\alpha \left(n\right)=1$ if $n$ is $N$-free ($N\ge 2$ a fixed integer), $\alpha \left(n\right)>1$ else, and $\alpha \left(2^n\right)\to \infty$. An asymptotic result is established with an error term probably definitive on the basis of the present knowledge about the zeros of the zeta-function. Applications to the enumerative functions of Abelian groups and of semisimple rings of given finite order are discussed.

We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in ${}_q\left[T\right]$ of degree d for which s consecutive coefficients $a=(a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a)={\mu }_{d}q+\left(q^{1/2}\right)$, where ${\mu }_d:={\sum }_{r=1}^d\left({(-1)}^{r-1}\right)/(r!)$. We also prove that $₂(d,s,a)=\mu {²}_{d}q²+\left(q^{3/2}\right)$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of ${}_q\left[T\right]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0)=\mu {²}_{d}q²+\left(q\right)$, where ₂(d,0) denotes the average second moment for all monic polynomials...

We investigate the value-distribution of Epstein zeta-functions ζ(s; Q), where Q is a positive definite quadratic form in n variables. We prove an asymptotic formula for the number of c-values, i.e., the roots of the equation ζ(s; Q) = c, where c is any fixed complex number. Moreover, we show that, in general, these c-values are asymmetrically distributed with respect to the critical line Re s =n/4. This complements previous results on the zero-distribution.[Proceedings of the Primeras Jornadas...

Let L be a finite Galois CM-extension of a totally real field K. We show that the validity of an appropriate special case of the Equivariant Tamagawa Number Conjecture leads to a natural construction for each odd prime p of explicit elements in the (non-commutative) Fitting invariants over ${\mathbb{Z}}_p\left[G\right]$ of a certain tame ray class group, and hence also in the analogous Fitting invariants of the p-primary part of the ideal class group of L. These elements involve the values at s=1 of the Artin L-series of characters...