On the constant in Burgess' bound for the number of consecutive residues or non-residues (Q449617)

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\[NEWLINEH<7.06p^{1/4}\log p\quad\text{for}\;p\geq 5\cdot 10^{18}.NEWLINE\]