On the constant in Burgess’ bound for the number of consecutive residues or non-residues. (English) Zbl 1301.11061
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:
\[ H<(\pi e \sqrt 6/3 +o(1))p^{1/4}\log p; \tag{1} \]
\[ H<7p^{1/4}\log p\quad\text{for}\;p\geq 5\cdot 10^{55}\tag{2} \] and \[ H<7.06p^{1/4}\log p\quad\text{for}\;p\geq 5\cdot 10^{18}. \]
\[ H<(\pi e \sqrt 6/3 +o(1))p^{1/4}\log p; \tag{1} \]
\[ H<7p^{1/4}\log p\quad\text{for}\;p\geq 5\cdot 10^{55}\tag{2} \] and \[ H<7.06p^{1/4}\log p\quad\text{for}\;p\geq 5\cdot 10^{18}. \]
Reviewer: Yong-Gao Chen (Nanjing)
MSC:
| 11L40 | Estimates on character sums |
| 11N25 | Distribution of integers with specified multiplicative constraints |
| 11L26 | Sums over arbitrary intervals |
| 11A15 | Power residues, reciprocity |
References:
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