SQUAREFULL NUMBERS IN ARITHMETIC PROGRESSIONS

2013 ◽  
Vol 09 (04) ◽  
pp. 885-901 ◽  
Author(s):  
TSZ HO CHAN ◽  
KAI MAN TSANG
Keyword(s):  
Character Sums ◽  
Arithmetic Progressions ◽  
Large Sieve ◽  
Large Sieve Inequality ◽  
Almost All

In this paper, we study squarefull numbers in arithmetic progressions. We find the least such squarefull number by Dirichlet's hyperbola method as well as Burgess bound on character sums. We also obtain a best possible almost all result via a large sieve inequality of Heath-Brown on real characters.

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].

Numbers badly approximate by fractions with prime denominator

1995 ◽  
Vol 118 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Continued Fraction ◽  
Arithmetic Progressions ◽  
The Other ◽  
Strong Result ◽  
Infinitely Many Solutions ◽  
Continued Fraction Expansion ◽  
Primes In Arithmetic Progressions ◽  
Prime Variable ◽  
Image Position ◽  
Almost All

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequalityhas infinitely many solutions if and only ifdiverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

On a Congruence Related to Character Sums

1985 ◽  
Vol 28 (4) ◽  
pp. 431-439 ◽  
Author(s):  
J. H. H. Chalk
Keyword(s):  
Prime Power ◽  
Dirichlet Character ◽  
Character Sums ◽  
Arithmetic Progressions ◽  
Solution Set ◽  
Character Sum ◽  
Equal Degree

AbstractIf χ is a Dirichlet character to a prime-power modulus pα, then the problem of estimating an incomplete character sum of the form ∑1≤x≤h χ (x) by the method of D. A. Burgess leads to a consideration of congruences of the typef(x)g'(x) - f'(x)g(x) ≡ 0(pα),where fg(x) ≢ 0(p) and f, g are monic polynomials of equal degree with coefficients in Ζ. Here, a characterization of the solution-set for cubics is given in terms of explicit arithmetic progressions.

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.

scholarly journals The variance of divisor sums in arithmetic progressions

2018 ◽  
Vol 30 (2) ◽  
pp. 269-293
Author(s):  
Brad Rodgers ◽  
Kannan Soundararajan
Keyword(s):  
Arithmetic Progressions ◽  
Large Sieve ◽  
Restricted Range ◽  
Divisor Function ◽  
Residue Classes ◽  
Divisor Sums

AbstractWe study the variance of sums of thek-fold divisor function{d_{k}(n)}over sparse arithmetic progressions, with averaging over both residue classes and moduli. In a restricted range, we confirm an averaged version of a recent conjecture about the asymptotics of this variance. This result is closely related to moments of DirichletL-functions, and our proof relies on the asymptotic large sieve.

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 Factorization invariants of the additive structure of exponential Puiseux semirings

2022 ◽  
Author(s):  
Harold Polo
Keyword(s):  
Rational Number ◽  
Arithmetic Progressions ◽  
Additive Structure ◽  
Positive Rational Number ◽  
Sets Of Lengths ◽  
Almost All

Exponential Puiseux semirings are additive submonoids of [Formula: see text] generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

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).

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