On a logarithmic sum related to a natural quadratic sieve

We study the sum $\Sigma_q(U)=\sum_{\substack{d,e\leq U\\(de,q)=1}}\frac{\mu(d)\mu(e)}{[d,e]}\log\left(\frac{U}{d}\right)\log\left(\frac{U}{e}\right)$, $U>1$, so that a continuous, monotonic and explicit version of Selberg's sieve can be stated. Thanks to Barban-Vehov (1968), Motohashi (1974) and Graham (1978), it has been long known, but never explicitly, that $\Sigma_1(U)$ is asymptotic to $\log(U)$. In this article, we discover not only that $\Sigma_q(U)\sim\frac{q}{\varphi(q)}\log(U)$ for all $q\in\mathbb{Z}_{>0}$, but also we find a closed-form expression for its secondary order term of $\Sigma_q(U)$, a constant $\mathfrak{s}_q$, which we are able to estimate explicitly when $q=v\in\{1,2\}$. We thus have $\Sigma_v(U)= \frac{v}{\varphi(v)}\log(U)-\mathfrak{s}_v+O_v^*\left(\frac{K_v}{\log(U)}\right)$, for some explicit constant $K_v > 0$, where $\mathfrak{s}_1=0.60731\ldots$ and $\mathfrak{s}_2=1.4728\ldots$. As an application, we show how our result gives an explicit version of the Brun-Titchmarsh theorem within a range.Comment: accepted in Acta Arithmetic

Explicit averages of square-free supported functions: to the edge of the convolution method

We give a general statement of the convolution method so that one can provide explicit asymptotic estimations for all averages of square-free supported arithmetic functions that have a sufficiently regular order on the prime numbers and observe how the nature of this method gives error term estimations of order $X^{-\delta}$, where $\delta$ belongs to an open real positive set $I$. In order to have a better error estimation, a natural question is whether or not we can achieve an error term of critical order $X^{-\delta_0}$, where $\delta_0$, the critical exponent, is the right hand endpoint of $I$. We reply positively to that question by presenting a new method that improves qualitatively almost all instances of the convolution method under some regularity conditions; now, the asymptotic estimation of averages of well-behaved square-free supported arithmetic functions can be given with its critical exponent and a reasonable explicit error constant. We illustrate this new method by analyzing a particular average related to the work of Ramar\'e--Akhilesh (2017), which leads to notable improvements when imposing non-trivial coprimality conditions.Comment: Updated. Some correction

Patient Access to Electronic Health Records: Strengths, weaknesses and what’s needed to move forward

Electronic health records (EHRs) are desired by both physicians and patients, but the transition to and acceptance of sensitive health information online has been slow. This paper reviews the current literature on EHR adoption and outlines barriers, advantages and explicit steps for moving toward the EHR ubiquity. Potential benefits of EHRs to patients and physicians include reduced costs for patients, hospitals and insurance providers, patient empowerment, less errors in records and better health outcomes, but security and privacy concerns, cost of implementation and poor electronic records management system design have proved barriers to adoption

A two end family of solutions for the Inhomogeneous Allen-Cahn equation in R^2

In this work we construct a family of entire bounded solution for the singulary perturbed Inhomogeneous Allen-Cahn Equation \ep^2\div\left(a(x)\nabla u\right)-a(x)F'(u)=0 in $\R^2$, where \ep\to 0. The nodal set of these solutions is close to a "nondegenerate" curve which is asymptotically two non paralell straight lines. Here $F'$ is a double-well potential and $a$ is a smooth positive function. We also provide example of curves and functions $a$ where our result applies. This work is in connection with the results found by Z.Du and B.Lai, Z.Du and C.Gui, and F. Pacard and M. Ritore, in "Transition layers for an inhomogeneus Allen-Cahn equation in Riemannian Manifolds", "Interior layers for an inhomogeneous Allen-Cahn equation", "From the constant mean curvature hypersurfaces to the gradient theory of phase transitions" respectively, where they handle the compact case.Comment: arXiv admin note: text overlap with arXiv:0902.2047 by other author

Fundamental solutions of pseudo-differential operators over p-adic fields

We show the existence of fundamental solutions for p-adic pseudo-differential operators with polynomial symbols.Comment: To appear in Rend. Sem. Mat. Univ. Padov

Exponential Sums Along p-adic Curves

Let K be a p-adic field, R the valuation ring of K, and P the maximal ideal of R. Let $Y subseteq R^{2}$ be a non-singular closed curve, and Y_{m} its image in R/P^{m} times R/P^{m}, i.e. the reduction modulo P^{m} of Y. We denote by Psi an standard additive character on K. In this paper we discuss the estimation of exponential sums of type S_{m}(z,Psi,Y,g):= sum\limits_{x in Y_{m}} Psi(zg(x)), with z in K, and g a polynomial function on Y. We show that if the p-adic absolute value of z is big enough then the complex absolute value of S_{m}(z,Psi,Y,g) is O(q^{m(1-beta(f,g))}), for a positive constant beta(f,g) satisfying 0<beta(f,g)<1.Comment: 9 pages. Accepted in Finite Fields and Their Application