Large Sieve Research Articles

Large sieve inequalities for periods of Maass forms

For Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} a Fuchsian Group of the first kind, we obtain large sieve inequalities with weights the hyperbolic periods of Maass forms of even weight. This is inspired by work of Chamizo, who proved a large sieve inequality with weights values of Maass forms of weight 0. The motivation is applications in counting problems in Γ1\\Γ/Γ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1 \\backslash \\Gamma /\\Gamma _2$$\\end{document}, where Γ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1$$\\end{document}, Γ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _2$$\\end{document} are hyperbolic subgroups of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}.

Log-free zero density estimates for automorphic L-functions

We prove a log-free zero density estimate for automorphic L-functions defined over a number field k. This work generalizes and sharpens the method of pseudo-characters and the large sieve used earlier by Kowalski and Michel. As applications, we demonstrate for a particular family of number fields of degree n over k (for any n) that an effective Chebotarev density theorem and a bound on ℓ-torsion in class groups hold for almost all fields in the family.

Oversampling and Donoho–Logan type theorems in model spaces

The aim of this paper is to extend two results from the Paley–Wiener setting to more general model spaces. The first one is an analogue of the oversampling Shannon sampling formula. The second one is a version of the Donoho–Logan Large Sieve Theorem which is a quantitative estimate of the embedding of the Paley–Wiener space into an \(L^2(\mathbb{R},\mu)\) space.

Exceptional biases in counting primes over function fields

AbstractWe study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call “complete bias,” “lower order bias,” and “reversed bias,” occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in as , a power of a fixed prime, goes to infinity. The bounds given extend a previous result of Kowalski, who studied a similar question along particular one‐parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret–Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes

Abstract We make explicit Bombieri’s refinement of Gallagher’s log-free “large sieve density estimate near σ = 1 {\sigma=1} ” for Dirichlet L-functions. We use this estimate and recent work of Green to prove that if N ≥ 2 {N\geq 2} is an integer, A ⊆ { 1 , … , N } {A\subseteq\{1,\ldots,N\}} , and for all primes p no two elements in A differ by p - 1 {p-1} , then | A | ≪ N 1 - 10 - 18 {|A|\ll N^{1-10^{-18}}} . This strengthens a theorem of Sárközy.

The large sieve revisited

The large sieve revisited

Poissonian Pair Correlation for $\alpha n^{\theta }$ mod 1

Abstract We show that sequences of the form $\alpha n^{\theta } \pmod {1}$ with $\alpha> 0$ and $0 < \theta < \tfrac {43}{117} = \tfrac {1}{3} + 0.0341 \ldots $ have Poissonian pair correlation. This improves upon the previous result by Lutsko, Sourmelidis, and Technau, where this was established for $\alpha> 0$ and $0 < \theta < \tfrac {14}{41} = \tfrac {1}{3} + 0.0081 \ldots $. We reduce the problem of establishing Poissonian pair correlation to a counting problem using a form of amplification and the Bombieri–Iwaniec double large sieve. The counting problem is then resolved non-optimally by appealing to the bounds of Robert–Sargos and (Fouvry–Iwaniec–)Cao–Zhai. The exponent $\theta = \tfrac {2}{5}$ is the limit of our approach.

Scarcity of congruences for the partition function

abstract: The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0$ $({\rm mod}\;\ell)$ for the primes $\ell=5,7,11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+\beta)\equiv 0$ $({\rm mod}\;\ell)$ where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.

A pretentious proof of Linnik's estimate for primes in arithmetic progressions

AbstractIn the present paper, the author adopts a pretentious approach and recovers an estimate obtained by Linnik for the sums of the von Mangoldt function Λ on arithmetic progressions. It is the analogue of an estimate that Linnik established in his attempt to prove his celebrated theorem concerning the size of the smallest prime number of an arithmetic progression. Our work builds on ideas coming from the pretentious large sieve of Granville, Harper, and Soundararajan and it also borrows insights from the treatment of Koukoulopoulos on multiplicative functions with small averages.

On the spectral large sieve inequality for symmetric-squares

Abstract We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.

Three-dimensional exponential sums under constant perturbation

By generalizing Bombieri and Iwaniec’s double large sieve inequality, we obtain an estimation on three-dimensional exponential sums with constant perturbation. As an application of our estimation on the three-dimensional exponential sums, we get that for any ϵ>0, ∑n⩽xΛ([xn])=x∑d⩾1Λ(d)d(d+1)+Oϵ(x1736+ϵ) as x→∞. This improves a recent result of Liu–Wu–Yang, which requires 9/19 in place of 17/36.

The large sieve with prime moduli

The large sieve type estimates of true order of magnitude for character sums to prime moduli are established. The main result holds for coefficients supported on numbers which have no small prime divisors.

Linnik's large sieve and the L1$L^{1}$ norm of exponential sums

AbstractThe elementary method of Balog and Ruzsa and the large sieve of Linnik are utilized to investigate the behaviour of the norm of an exponential sum over the primes. A new proof of a lower bound due to Vaughan for the norm of an exponential sum formed with the von Mangoldt function is furnished.

The large sieve with power moduli in imaginary quadratic number fields

In this paper, we establish large sieve inequalities for power moduli in imaginary quadratic number fields, extending earlier work of Baier and Bansal [S. Baier and A. Bansal, The large sieve with power moduli for [Formula: see text], Int. J. Number Theory 14 (10) (2018) 2737–2756; Large sieve with sparse sets of moduli for [Formula: see text], Acta Arith. 196 (1) (2020) 17–34] for the Gaussian field.

Additive energy and a large sieve inequality for sparse sequences

We consider the large sieve inequality for sparse sequences of moduli and give a general result depending on the additive energy (both symmetric and asymmetric) of the sequence of moduli. For example, in the case of monomials f ( X ) = X k $f(X) = X^k$ this allows us to improve, in some ranges of the parameters, the previous bounds of S. Baier and L. Zhao (2005), K. Halupczok (2012, 2015, 2018) and M. Munsch (2020). We also consider moduli defined by polynomials f ( X ) ∈ Z [ X ] $f(X) \in \mathbb {Z}[X]$ , Piatetski–Shapiro sequences and general convex sequences. We then apply our results to obtain a version of the Bombieri–Vinogradov theorem with Piatetski–Shapiro moduli improving the level of distribution of R. C. Baker (2014).

A problem in comparative order theory

Write \(\mathrm {ord}_p(\cdot )\) for the multiplicative order in \({\mathbb {F}}_p^{\times }\). Recently, Matthew Just and the second author investigated the problem of classifying pairs \(\alpha , \beta \in {\mathbb {Q}}^{\times }\setminus \{\pm 1\}\) for which \(\mathrm {ord}_p(\alpha ) > \mathrm {ord}_p(\beta )\) holds for infinitely many primes p. They called such pairs order-dominant. We describe an easily-checkable sufficient condition for \(\alpha ,\beta \) to be order-dominant. Via the large sieve, we show that almost all integer pairs \(\alpha ,\beta \) satisfy our condition, with a power savings on the size of the exceptional set.

Similarities in Evolution of Aggregate Size Distributions during Successive Wetting and Drying Cycles of Heavy Textured Soils of Variable Clay Mineralogy

A phenomenon causing instability of soil structure and associated hydraulic properties in recently tilled soils is aggregate fragmentation induced by wetting and drying cycles. We analyzed data from three experiments in Puerto Rico, the UK and China measuring fragmentation and resulting evolution of aggregate size distributions during successive wetting and drying cycles in heavy textured soils. Aggregate distributions were represented as the cumulative fraction F of aggregates passing through successively larger sieve sizes X. To a good approximation, all distributions exhibited similarity in that the aggregate diameter X(F) corresponding to F in a given test distribution was always a characteristic multiple α¯ of X(F) in a fixed reference distribution, where α¯ for a distribution was calculated as its mean weight aggregate diameter (MWD) divided by the MWD of the reference distribution. In most cases, α¯ for a given soil varied inversely with the square of the number of wetting and drying cycles. For different soils of similar initial aggregate sizes, α¯ for a given wet–dry cycle decreased with increasing activity coefficient, reflecting the enhancing effect of soil shrink–swell potential on fragmentation. Results highlight usefulness of the van Bavel mean weight diameter as a natural scaling parameter for characterizing aggregate distributions.

Some inequalities of the large sieve type

Some inequalities of the large sieve type

An improved spectral large sieve inequality for ${\rm SL}_3(\mathbb {Z})$

We prove an improved spectral large sieve inequality for the family of ${\rm SL }_3(\mathbb {Z})$ Hecke–Maass cusp forms. The method of proof uses duality, and its structure reveals unexpected connections to Heath-Brown’s large sieve for cubic characters.

Large sieve estimate for multivariate polynomial moduli and applications

We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri–Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens previous results of the first author in two aspects: the range of the moduli as well as the class of polynomials which can be handled. As a consequence, we deduce that there exist infinitely many primes p such that \(p-1\) has a prime divisor of size \(\gg p^{2/5+o(1)}\) that is the value of an incomplete norm form polynomial.