arXiv subjects
Publications and source records attributed to Giuseppe Molteni.
We provide explicit upper and lower bounds for certain commonly occurring expressions that involve the Riemann zeta function $ζ(s)$ and its derivatives on $s>1$. For example, this improves upon existing bounds for $(s-1)ζ(s)$ and its derivatives, and for $(ζ'/ζ)(s)$.
We prove new explicit conditional bounds for the residue at $s=1$ of the Dedekind zeta-function associated to a number field. Our bounds are concrete and all constants are presented with explicit numerical values.
An Egyptian fraction is a sum of the form $1/n_1 + \cdots + 1/n_r$ where $n_1, \dots, n_k$ are distinct positive integers. We prove explicit lower bounds for the cardinality of the set $E_N$ of rational numbers that can be represented by Egyptian fractions with denominators not exceeding $N$. More precisely, we show that for every integer $k \geq 4$ such that $\ln_k N \geq 3/2$ it holds $$ \frac{\ln(|E_N|)}{\ln 2} \geq \Big(2 - \frac{3}{\ln_k N}\Big)\frac{N}{\ln N}\prod_{j=3}^{k} \ln_j N , $$ where $\ln_k$ denotes the $k$-th iterate of the natural logarithm. This improves on a previous result of Bleicher and Erdős who established a similar bound but under the more stringent condition $\ln_k N\geq k$ and with a leading constant of $1$. Furthermore, we provide some methods to compute the exact values of $|E_N|$ for large positive integers $N$, and we give a table of $|E_N|$ for $N$ up to $154$.
Let $K$ be a field of degree $n$ and discriminant with absolute value $Δ$. Under the assumption of the validity of the Generalized Riemann Hypothesis, we provide a new algorithm to compute a set of generators of the class group of $K$ and prove that the norm of the ideals in that set is $\leq (4-1/(2n))\log^2Δ$, except for a finite number of fields of degree $n\leq 4$. For those fields, the conclusion holds with the slightly larger limit $(4-1/(2n)+1/(2n^2))\log^2Δ$. When the cardinality of $\mathcal C\!\ell$ is odd the bounds improve to $(4-2/(3n))\log^2Δ$, again with finitely many exceptions in degree $n\leq 4$, and to $(4-2/(3n)+3/(8n^2))\log^2Δ$ without exceptions.
For every $α\in (0,+\infty)$ and $p,q \in (1,+\infty)$ let $T_α$ be the operator $L^p[0,1]\to L^q[0,1]$ defined via the equality $(T_αf)(x) := \int_0^{x^α} f(y) d y$. We study the norms of $T_α$ for every $p$, $q$. In the case $p=q$ we further study its spectrum, point spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case $p=q=2$ we determine the point spectrum and eigenfunctions for $T^*_αT_α$, where $T^*_α$ is the adjoint operator.
Let $d$ be any positive and non square integer. We prove an upper bound for the first two moments of the length $T(d)$ of the period of the continued fraction expansion for $\sqrt{d}$. This allows to improve the existing results for the large deviations of $T(d)$.
Current methods for the classification of number fields with small regulator depend mainly on an upper bound for the discriminant, which can be improved by looking for the best possible upper bound of a specific polynomial function over an hypercube. In this paper, we provide new and effective upper bounds for the case of fields with one complex embedding and degree between five and nine: this is done by adapting the strategy we have adopted to study the totally real case, but for this new setting several new computational issues had to be overcome. As a consequence, we detect the four number fields of signature (6,1) with smallest regulator; we also expand current lists of number fields with small regulator in signatures (3,1), (4,1) and (5,1).
Pattern classification with compact representation is an important component in machine intelligence. In this work, an analytic bridge solution is proposed for compressive classification. The proposal has been based upon solving a penalized error formulation utilizing an approximated $\ell_p$-norm. The solution comes in a primal form for over-determined systems and in a dual form for under-determined systems. While the primal form is suitable for problems of low dimension with large data samples, the dual form is suitable for problems of high dimension but with a small number of data samples. The solution has also been extended for problems with multiple classification outputs. Numerical studies based on simulated and real-world data validated the effectiveness of the proposed solution.
Let $P_n(y_1,\ldots,y_n):= \prod_{1\leq i<j\leq n}\left( 1 -\frac{y_i}{y_j}\right) $ and $P_n:= \sup_{(y_1,\ldots,y_n)}P_n(y_1,\ldots,y_n) $ where the supremum is taken over the $n$-ples $(y_1,\ldots,y_n)$ of real numbers satisfying $0 <|y_1| < |y_2|< \cdots < |y_n|$. We prove that $P_n \leq 2^{\lfloor n/2\rfloor}$ for every $n$, i.e., we extend to all $n$ the bound that Pohst proved for $n\leq 11$. As a consequence, the bound for the absolute discriminant of a totally real field in terms of its regulator is now proved for every degree of the field.
Given a real number $τ$, we study the approximation of $τ$ by signed harmonic sums $σ_N(τ) := \sum_{n \leq N}{s_n(τ)}/n$, where the sequence of signs $(s_N(τ))_{N \in\mathbb{N}}$ is defined "greedily" by setting $s_{N+1}(τ) := +1$ if $σ_N(τ) \leq τ$, and $s_{N+1}(τ) := -1$ otherwise. Precisely, we compute the limit points and the decay rate of the sequence $(σ_N(τ)-τ)_{N \in \mathbb{N}}$. Moreover, we give an accurate description of the behavior of the sequence of signs $(s_N(τ))_{N\in\mathbb{N}}$, highlighting a surprising connection with the Thue--Morse sequence.
We prove an explicit upper bound for the k-th prime ideal with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions.
For any integer $N \geq 1$, let $\mathfrak{E}_N$ be the set of all Egyptian fractions employing denominators less than or equal to $N$. We give upper and lower bounds for the cardinality of $\mathfrak{E}_N$, proving that $$ \frac{N}{\log N} \prod_{j = 3}^{k} \log_j N<\log(\#\mathfrak{E}_N) < 0.421\, N, $$ for any fixed integer $k\geq 3$ and every sufficiently large $N$, where $\log_j x$ denotes the $j$-th iterated logarithm of $x$.
We prove explicit versions of Cramér's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.
For every $τ\in\mathbb{R}$ and every integer $N$, let $\mathfrak{m}_N(τ)$ be the minimum of the distance of $τ$ from the sums $\sum_{n=1}^N s_n/n$, where $s_1, \ldots, s_n \in \{-1, +1\}$. We prove that $\mathfrak{m}_N(τ) < \exp\!\big(-C(\log N)^2\big)$, for all sufficiently large positive integers $N$ (depending on $C$ and $τ$), where $C$ is any positive constant less than $1/\log 4$.
We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.
Assuming Generalized Riemann's Hypothesis, Bach proved that the class group $\mathcal C\!\ell_{\mathbf K}$ of a number field ${\mathbf K}$ may be generated using prime ideals whose norm is bounded by $12\log^2Δ_{\mathbf K}$, and by $(4+o(1))\log^2Δ_{\mathbf K}$ asymptotically, where $Δ_{\mathbf K}$ is the absolute value of the discriminant of ${\mathbf K}$. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates $\mathcal C\!\ell_{\mathbf K}$ and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that $\mathcal C\!\ell_{\mathbf K}$ is generated by prime ideals whose norm is bounded by the minimum of $4.01\log^2Δ_{\mathbf K}$, $4\big(1+\big(2πe^γ)^{-n_{\mathbf K}}\big)^2\log^2Δ_{\mathbf K}$ and $4\big(\logΔ_{\mathbf K}+\log\logΔ_{\mathbf K}-(γ+\log 2π)n_{\mathbf K}+1+(n_{\mathbf K}+1)\frac{\log(7\logΔ_{\mathbf K})}{\logΔ_{\mathbf K}}\big)^2$. Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size $\asymp (\logΔ_{\mathbf K}\log\logΔ_{\mathbf K})^2$. In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be experimentally smaller than $\log^2Δ_{\mathbf K}$ except for 7 out of 31000 fields.
We have proved recently several explicit versions of the prime ideal theorem under GRH. Here we prove a version with optimal asymptotic behaviour.
In [BDyDF08] Belabas, Diaz y Diaz and Friedman show a way to determine, assuming the Generalized Riemann Hypothesis, a set of prime ideals that generate the class group of a number field. Their method is efficient because it produces a set of ideals that is smaller than earlier proved results. Here we show how to use their main result to algorithmically produce a bound that is lower than the one they prove.