Large values of character sums

Hildebrand, Adolf

doi:10.1016/0022-314x(88)90106-0

Cited by 28 publications

(29 citation statements)

References 5 publications

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“…which is valid for all functions f with |f (n)| ≤ 1, and all 1 ≤ y ≤ √ x. Using this estimate (in place of Proposition 4.1) and arguing exactly as in the proof of Proposition 4.5 we arrive at the following Proposition (see also Lemma 5 of Hildebrand [11]). where s(f, y) = p≤y |1 − f (p)|/p.…”

Section: Generalized Notions Of the Spectrum: The Logarithmic Spectrummentioning

confidence: 64%

The Spectrum of Multiplicative Functions

Granville¹,

Soundararajan²

2001

The Annals of Mathematics

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Section: Generalized Notions Of the Spectrum: The Logarithmic Spectrummentioning

confidence: 64%

The Spectrum of Multiplicative Functions

Granville¹,

Soundararajan²

2001

The Annals of Mathematics

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“…Littlewood's result (1.6) indicates that L(1, χ) is large only when χ(p) ≈ 1 for many small primes p. We will find that the other terms appearing in (2.1) can be large only when χ(p) ≈ ξ(p) for many small primes p, where ξ is a character of small conductor. A. Hildebrand [10] first realized the possibility of such a result.…”

Section: Detailed Statement Of Resultsmentioning

confidence: 99%

“…In the final section of this paper we use the improved upper bounds for L(1, χ) given in [9] to obtain a modest improvement over Hildebrand's results [10] on the constant in the Pólya-Vinogradov inequality. …”

Section: Conjecture 1 If χ Is a Primitive Charactermentioning

confidence: 99%

Large character sums: Pretentious characters and the Pólya-Vinogradov theorem

Granville¹,

Soundararajan

2006

J. Amer. Math. Soc.

117

192

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In 1918 Pólya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Pólya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Pólya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters.

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“…We make one final cosmetic adjustment prior to estimating this quantity. Hildebrand proved the following useful result (see Lemma 5 of [7]): for any g ∈ F and x ≥ 1,…”

Section: The Major Arc Case: Proof Of Theorem 27mentioning

confidence: 99%

Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality

Goldmakher

2012

ANT

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We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd-order character sums. Furthermore, conditionally on the generalized Riemann hypothesis we obtain a bound for odd-order character sums which is best possible.

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Large values of character sums

Cited by 28 publications

References 5 publications

The Spectrum of Multiplicative Functions

The Spectrum of Multiplicative Functions

Large character sums: Pretentious characters and the Pólya-Vinogradov theorem

Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality

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