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A note on some results of Hua in short intervals

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Abstract

In this paper, we improve the previous results of the authors [G. Lü and H. Tang, On some results of Hua in short intervals, Lith. Math. J., 50(1):54–70, 2010] by proving that each sufficiently large integer N satisfying some congruence conditions can be written as

$$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k},} \hfill \\ {\left| {{p_j} - \sqrt {{\frac{N}{5}}} } \right| \leqslant U,\quad \left| {p - {{\left( {\frac{N}{5}} \right)}^{\frac{1}{k}}}} \right| \leqslant U\,{N^{ - \frac{1}{2} + \frac{1}{k}}},\quad j = 1,\,2,\,\,3,\,4,} \hfill \\ \end{array} } \right. $$

where U = N 1/2−η+ε with \( \eta = \frac{1}{{2k\left( {{K^2} + 1} \right)}} \) and K = 2k −1, k ⩾ 2.

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  1. School of Mathematics and Information Sciences, Henan University, Kaifeng, Henan, 475001, PR China

    Hengcai Tang

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  1. Hengcai Tang
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Tang, H. A note on some results of Hua in short intervals. Lith Math J 51, 75–81 (2011). https://doi.org/10.1007/s10986-011-9109-5

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  • DOI: https://doi.org/10.1007/s10986-011-9109-5

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