Large character sums

Granville, Andrew; Soundararajan, K.

doi:10.1090/s0894-0347-00-00357-x

Cited by 72 publications

(91 citation statements)

References 13 publications

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“…Our proofs of Theorems 1 and 2 are by the method of moments. Unfortunately, Granville and Soundrarajan [2] have shown that the fourth moment of the full sum (1•1) does not converge as x → ∞, so that this method yields no information about the distribution of (1•1). Following our proofs of Theorems 1 and 2 we present the results of some Monte Carlo experiments concerning the whole sum (restricted to squarefree numbers), which seem to suggest that its limiting distribution is not Gaussian.…”

Section: Bob Houghmentioning

confidence: 99%

“…Indeed, the inequality may be checked by interchanging the order of the iterated integrals. When t j is the innermost integral its range of integration is from 3/2 to some function of the other variables, and on such a range the integral against dπ(t j ) is majorized by the integral against the smooth measure dt j /log t j + αdt j /(log t j ) 2 . Switching measures one at a time we make a sequence of changes that always increases the result.…”

Section: Main Lemmamentioning

confidence: 99%

“…Summation of a random multiplicative function 211 with the a(n) bounded by 2k k , and so, denoting 6 s(n) (2) |{n 2 ∈ P k (x, x + y) (2) :…”

Section: An Upper Bound For the Law Of The Iterated Logarithmmentioning

confidence: 99%

“…Thus the = 1 term contributes a main term of |P k (x, x + y)| 2 k |P k (y)| 2 . For the > 1 terms, note that the summand is 0 unless ∈ P 2 j for some 1 j k, since all elements of P k (x, x + y) (2) are composed of exactly 2k primes. Thus n ∈ P k (x, x + y) (2) : s(n) = =…”

Section: An Upper Bound For the Law Of The Iterated Logarithmmentioning

confidence: 99%

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Summation of a random multiplicative function on numbers having few prime factors

Hough¹

2010

Math. Proc. Camb. Phil. Soc.

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Given a ±1 random completely multiplicative function f, we prove by estimating moments that the limiting distribution of the normalized sum converges to the standard Gaussian distribution as x → ∞ when r restricts summation to n having o(log log log x) prime factors. We also give an upper bound for the large deviations of with the sum restricted to numbers having a fixed number k of prime factors.

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Section: Bob Houghmentioning

confidence: 99%

Section: Main Lemmamentioning

confidence: 99%

“…Summation of a random multiplicative function 211 with the a(n) bounded by 2k k , and so, denoting 6 s(n) (2) |{n 2 ∈ P k (x, x + y) (2) :…”

Section: An Upper Bound For the Law Of The Iterated Logarithmmentioning

confidence: 99%

Section: An Upper Bound For the Law Of The Iterated Logarithmmentioning

confidence: 99%

See 2 more Smart Citations

Summation of a random multiplicative function on numbers having few prime factors

Hough¹

2010

Math. Proc. Camb. Phil. Soc.

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“…Dans , améliorant des estimations de , Hough établit une minoration de la quantité

Δ false(x, q false) : = \underset{\frac{0pt}{χ \neq χ_{0}}}{trueprefixmax} |S false(x, χ false)|,

lorsque

q

est un nombre premier. Notant

x = q^{ϑ}

, cette estimation est valide dans le domaine

4 \sqrt{\frac{\log_{2} q}{\log q}} \log_{3} q ⩽ ϑ ⩽ 1 - 4 \sqrt{\frac{\log_{2} q}{\log q}} \log_{3} q .

…”

Section: Introduction Et éNoncé Des Résultatsunclassified

Sommes de Gál et applications

Bretèche

Tenenbaum

2018

Proc. London Math. Soc.

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We evaluate the asymptotic size of various sums of Gál type, in particular 0trueS(M):=∑m,n∈scriptM(m,n)[m,n],where scriptM is a finite set of integers. Elaborating on methods recently developed by Bondarenko and Seip, we obtain an asymptotic formula for log supfalse|scriptMfalse|=NS(M)/Nand derive new lower bounds for localized extreme values of the Riemann zeta‐function, for extremal values of some Dirichlet L‐functions at s=12, and for large character sums.

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Running Time Predictions for Factoring Algorithms

Croot

Granville

Pemantle

et al. 2008

Lecture Notes in Computer Science

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In 1994, Carl Pomerance proposed the following problem: Select integers a 1 , a 2 , . . . , a J at random from the interval [1, x], stopping when some (non-empty) subsequence, {a i : i ∈ I} where I ⊆ {1, 2, . . . , J}, has a square product (that is i∈I a i ∈ Z 2 ). What can we say about the possible stopping times, J? A 1985 algorithm of Schroeppel can be used to show that this process stops after selecting (1 + )J 0 (x) integers a j with probability 1 − o(1) (where the function J 0 (x) is given explicitly in (1) below). Schroeppel's algorithm actually finds the square product, and this has subsequently been adopted, with relatively minor modifications, by all factorers. In 1994 Pomerance showed that, with probability 1−o(1), the process will run through at least J 0 (x) 1−o(1) integers a j , and asked for a more precise estimate of the stopping time. We conjecture that there is a "sharp threshold" for this stopping time, that is, with probability 1 − o(1) one will first obtain a square product when (precisely) {e −γ + o(1)}J 0 (x) integers have been selected. Herein we will give a heuristic to justify our belief in this sharp transition.In our paper [4] we prove that, with probability 1 − o(1), the first square product appears in timewhere γ = 0.577... is the Euler-Mascheroni constant, improving both Schroeppel and Pomerance's results. In this article we will prove a weak version of this theorem (though still improving on the results of both Schroeppel and Pomerance). We also confirm the well established belief that, typically, none of the integers in the square product have large prime factors. Our methods provide an appropriate combinatorial framework for studying the large prime variations associated with the quadratic sieve and other factoring algorithms. This allows us to analyze what factorers have discovered in practice.

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Large character sums

Abstract: We make conjectures and give estimates for how large character sums can be as we vary over all characters mod q q , and as we vary over real, quadratic characters. In particular we show that the largest sums seem to depend on the value of the character at “smooth numbers”.

Cited by 72 publications

References 13 publications

Summation of a random multiplicative function on numbers having few prime factors

Summation of a random multiplicative function on numbers having few prime factors

Sommes de Gál et applications

Running Time Predictions for Factoring Algorithms

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