We study the sum $Σ_q(U)=\sum_{\substack{d,e\leq U\\(de,q)=1}}\frac{μ(d)μ(e)}{[d,e]}\log\left(\frac{U}{d}\right)\log\left(\frac{U}{e}\right)$, $U>1$, 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 $Σ_1(U)$ is asymptotic to $\log(U)$. In this article, we discover not only that $Σ_q(U)\sim\frac{q}{φ(q)}\log(U)$ for all $q\in\mathbb{Z}_{>0}$, but also we find a closed-form expression for its secondary order term of $Σ_q(U)$, a constant $\mathfrak{s}_q$, which we are able to estimate explicitly when $q=v\in\{1,2\}$. We thus have $Σ_v(U)= \frac{v}{φ(v)}\log(U)-\mathfrak{s}_v+O_v^*\left(\frac{K_v}{\log(U)}\right)$, 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.
We establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal for the Gaussian field.
In this paper, we establish a version of the large sieve with square moduli for imaginary quadratic extensions of rational function fields of odd characteristics.
We study the two-parameter quadratic sieve for a general test function. We prove, under some very general assumptions, that the function considered by Barban and Vehov [BV68] and Graham [Gra78] for this problem is optimal up to and including the second-order term. We determine that second-order term explicitly.
We develop a version of Ekedahl's geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.
Large sieve inequality for sums of Legendre symbols over short intervals
Université Jean Monnet, Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, ICJ UMR5208（让·莫内大学、里昂中央理工学院、里昂国立应用科学学院、里昂第一大学、法国国家科学研究中心、ICJ UMR5208）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, University of Bristol（布里斯托大学数学系）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
Motivated by a desire to understand the distribution of roots of cubic congruences, we re-derive a parametrization of roots $ν\pmod m$ of $X^3 \equiv 2 \pmod m$ found by Hooley. Although this parametrization does not lead us here to anything towards proving equidistribution of the sequence $\fracν{m} \pmod 1$, we are able to prove spacing results, and then a large sieve type inequality, which we view as analogous to the large sieve inequality for roots of quadratic congruences found by Fouvry and Iwaniec in their proof that there are infinitely many primes of the form $n^2 + p^2$. The parametr…
Let $d\in\mathbb{N}$, and let $π$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L'}{L}(1+it,π\otimesχ_D)$ as $D$ varies over fundamental discriminants. Here, $t$ is a fixed real number and $χ_D$ is the real character associated with $D$. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L'}{L}…
We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many sign patterns which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the $κ-1$-Fourier …
