The Large Sieve Inequality for the Exponential Sequence λ[O(n15/14+o(1))] Modulo Primes

2009 ◽  
Vol 61 (2) ◽  
pp. 336-350 ◽  
Author(s):  
M. Z. Garaev
Keyword(s):  
Large Sieve ◽  
Trigonometric Sums ◽  
Fixed Integer ◽  
Large Sieve Inequality ◽  
Positive Integers ◽  
Exponential Sequence

Abstract. Let ƛ be a fixed integer exceeding 1 and sn any strictly increasing sequence of positive integers satisfying sn ≤ n15/14+o(1). In this paper we give a version of the large sieve inequality for the sequence ƛsn. In particular, we obtain nontrivial estimates of the associated trigonometric sums “on average” and establish equidistribution properties of the numbers ƛsn, n ≤ p(log p)2+ϵ, modulo p for most primes p.

scholarly journals LARGE SIEVE INEQUALITY WITH CHARACTERS FOR POWERFUL MODULI

2005 ◽  
Vol 01 (02) ◽  
pp. 265-279 ◽  
Author(s):  
STEPHAN BAIER ◽  
LIANGYI ZHAO
Keyword(s):  
General Formula ◽  
Large Sieve ◽  
Large Sieve Inequality

In this paper we aim to generalize the results in [1, 2, 19] and develop a general formula for large sieve with characters to powerful moduli that will be an improvement to the result in [19].

scholarly journals Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Vincenzo De Filippis ◽  
Nadeem UR Rehman ◽  
Abu Zaid Ansari
Keyword(s):  
Lie Ideal ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Free Ring ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers ◽  
Torsion Free

LetRbe a 2-torsion free ring and letLbe a noncentral Lie ideal ofR, and letF:R→RandG:R→Rbe two generalized derivations ofR. We will analyse the structure ofRin the following cases: (a)Ris prime andF(um)=G(un)for allu∈Land fixed positive integersm≠n; (b)Ris prime andF((upvq)m)=G((vrus)n)for allu,v∈Land fixed integersm,n,p,q,r,s≥1; (c)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈[R,R]and fixed integern≥1; and (d)Ris semiprime andF((uv)n)=G((vu)n)for allu,v∈Rand fixed integern≥1.

On the k-th difference of an additive representation function

2011 ◽  
Vol 48 (1) ◽  
pp. 93-103
Author(s):  
Sándor Kiss
Keyword(s):  
Infinite Sequence ◽  
Probabilistic Methods ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Representation Function ◽  
Additive Representation ◽  
Consecutive Integers ◽  
Additive Representation Function ◽  
Positive Integers

Let k ≧ 2 be a fixed integer, A = {a1, a2, …} (a1 < a2 < …) be an infinite sequence of positive integers, and let Rk(n) denote the number of solutions of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$a_{i_1 } + a_{i_2 } + \cdots + a_{i_k } = n,a_{i_1 } \in \mathcal{A},...,a_{i_k } \in \mathcal{A}$$ \end{document}. Let B(A, N) denote the number of blocks formed by consecutive integers in A up to N. In [5], it was proved that if k > 2 and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\lim _{N \to \infty } \frac{{B(\mathcal{A},N)}}{{\sqrt[k]{N}}}$$ \end{document} = ∞ then |δl(Rk(n))| cannot be bounded for l ≦ k. The aim of this paper is to show that the above result is nearly best possible. We are using probabilistic methods.

scholarly journals A NEW BOUND FOR THE LARGE SIEVE INEQUALITY WITH POWER MODULI

2012 ◽  
Vol 08 (03) ◽  
pp. 689-695 ◽  
Author(s):  
KARIN HALUPCZOK
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Large Sieve ◽  
Large Sieve Inequality

We give a new bound for the large sieve inequality with power moduli qk that is uniform in k. The proof uses a new theorem due to Wooley from his work [Vinogradov's mean value theorem via efficient congruencing, to appear in Ann. of Math.] on efficient congruencing.

The large sieve with power moduli for ℤ[i]

2018 ◽  
Vol 14 (10) ◽  
pp. 2737-2756
Author(s):  
Stephan Baier ◽  
Arpit Bansal
Keyword(s):  
Classical Case ◽  
Large Sieve ◽  
Counting Problem ◽  
Large Sieve Inequality ◽  
Poisson Summation

We establish a large sieve inequality for power moduli in [Formula: see text], extending earlier work by Zhao and the first-named author on the large sieve for power moduli for the classical case of moduli in [Formula: see text]. Our method starts with a version of the large sieve for [Formula: see text]. We convert the resulting counting problem back into one for [Formula: see text] which we then attack using Weyl differencing and Poisson summation.

scholarly journals Global decomposition of GL(3) Kloosterman sums and the spectral large sieve

2019 ◽  
Vol 2019 (757) ◽  
pp. 51-88 ◽  
Author(s):  
Valentin Blomer ◽  
Jack Buttcane
Keyword(s):  
Kloosterman Sums ◽  
Bilinear Forms ◽  
Large Sieve ◽  
Large Sieve Inequality

AbstractWe prove best-possible bounds for bilinear forms in Kloosterman sums for \operatorname{GL}(3) associated with the long Weyl element. As an application we derive a best-possible spectral large sieve inequality on \operatorname{GL}(3).

scholarly journals Some extensions of additive properties of general sequences

2006 ◽  
Vol 73 (1) ◽  
pp. 139-146 ◽  
Author(s):  
Min Tang
Keyword(s):  
Infinite Sequence ◽  
Monotonicity Property ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
General Sequences ◽  
Positive Integers

Let A = {a1, a2,…}(a1 < a2 < …) be an infinite sequence of positive integers. Let k ≥ 2 be a fixed integer and denote by Rk(n) the number of solutions of . Erdős, Sárközy and Sós studied the boundness of |R2(n + 1) − R2(n)| and the monotonicity property of R2(n). In this paper, we extend some results to k > 2.

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