Analytic Number Theory

Iwaniec, Henryk; Kowalski, Emmanuel

doi:10.1090/coll/053

Cited by 1,724 publications

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“…Sums over consecutive integers. Following the usual approach to estimating incomplete sums (see [18,Section 12.2]), we first estimate the complete sums S a,b = n∈Z * t e p (aϑ 1/n )e t (bn).…”

Section: Preparationsmentioning

confidence: 99%

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Distribution of consecutive modular roots of an integer

Bourgain

Shparlinski

2008

Acta Arith.

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1. Introduction. For a prime p we denote by F p the finite field of p elements, which we assume to be represented by the set {0, 1, . . . , p − 1}. For an integer t we denote by Z t the residue ring modulo t and by Z * t the group of units of Z t .Let ϑ ∈ F * p be of multiplicative order t ≥ 1. There is a rather long history of studying the exponential sums with the powers ϑ n , n = M +1, . . . , M +N , where N ≤ t, in finite fields and residue rings: see [6,7,17,[21][22][23][26][27][28] for several results in this direction together with their numerous applications.Here we consider a presumably harder question about exponential sums with the roots ϑ 1/n , for n = M + 1, . . . , M + N with gcd(n, t) = 1 instead of the powers. More precisely, for an integer m > 0, we put

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Section: Preparationsmentioning

confidence: 99%

“…The standard technique (see [18,Section 12.2]) now immediately leads us to the following estimate of the sums S a (M, N ).…”

Section: Preparationsmentioning

confidence: 99%

Distribution of consecutive modular roots of an integer

Bourgain

Shparlinski

2008

Acta Arith.

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“…where G(u) is any function which is holomorphic in the strip −4 < (u) < 4, even and normalized by G(0) = 1 (see Theorem 5.3. in [6]). Let…”

Section: The Error Termmentioning

confidence: 98%

“…(This follows from the exponential decay of the function W t (see [6]). ) Our job is now reduced to producing a decent upper bound for U * (N, t).…”

Section: The Error Termmentioning

confidence: 99%

On quadratic families of CM elliptic curves

Munshi

2011

Trans. Amer. Math. Soc.

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Abstract. Given a CM elliptic curve with Weierstrass equation y 2 = f (x), and a positive definite binary quadratic form Q(u, v), we show that there are infinitely many reduced integer pairs (u, v) such that the twisted elliptic curve Q(u, v)y 2 = f (x) has analytic rank (and consequently Mordell-Weil rank) one. In fact it follows that the number of such pairs with |u|, |v| ≤ X is at least X 2−ε for any ε > 0.

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“…[IK,12.2]) leads to complete sums (i.e., with ϑ = 1), for which the results of algebraic geometry can often by applied, giving a bound of size roughly √ p(log p), so an estimate for S p,ϑ (g) with saving θ(N ) = N 1−1/(2ϑ) → +∞ (see (2)).…”

Section: Application: Definable Intervalsmentioning

confidence: 99%

Exponential sums over definable subsets of finite fields

Kowalski

2007

Isr. J. Math.

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Abstract. We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates over varieties over finite fields due to Weil, Deligne and others, and the result of Chatzidakis, van den Dries and Macintyre concerning the number of points of those definable sets. As a first application, there is no formula in the language of rings that defines for infinitely many primes an "interval" in Z/pZ that is neither bounded nor with bounded complement.

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Analytic Number Theory

Cited by 1,724 publications

References 0 publications

Distribution of consecutive modular roots of an integer

Distribution of consecutive modular roots of an integer

On quadratic families of CM elliptic curves

Exponential sums over definable subsets of finite fields

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