On the constant in Burgess' bound for the number of consecutive residues or non-residues (Q449617)
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scientific article; zbMATH DE number 6074844
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the constant in Burgess' bound for the number of consecutive residues or non-residues |
scientific article; zbMATH DE number 6074844 |
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On the constant in Burgess' bound for the number of consecutive residues or non-residues (English)
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31 August 2012
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Dirichlet character
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consecutive nonresidues
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power residues
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0.8592105
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0.8238554
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0.82342327
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0.8175311
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0.8112557
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0.80921394
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0.8063575
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0.8034717
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Let \(\chi\) be a non-principal Dirichlet character to the prime modulus \(p\) which is constant on \((N, N+H]\). A well known result due to D. A. Burgess is \(H=O(p^{1/4}\log p)\). In this paper, the author gives the quantitative versions. The following results are proved:NEWLINENEWLINE NEWLINE\[NEWLINEH<(\pi e \sqrt 6/3 +o(1))p^{1/4}\log p; \tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEH<7p^{1/4}\log p\quad\text{for}\;p\geq 5\cdot 10^{55}\tag{2}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEH<7.06p^{1/4}\log p\quad\text{for}\;p\geq 5\cdot 10^{18}.NEWLINE\]
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