Sums of nine almost equal prime cubes

Abstract

We prove that each sufficiently large odd integer N can be written as sum of the form N=p_{\mathrm{1}}^{\mathrm{3}}+p_{\mathrm{2}}^{\mathrm{3}}+\cdots +p_{\mathrm{9}}^{\mathrm{3}} with \left|p_j-{(N/\mathrm{9})}^{\mathrm{1}/\mathrm{3}}\right|\le N^{(\mathrm{1}/\mathrm{3})-\theta }, where p_j, j = 1, 2, … , 9, are primes and \theta =(\mathrm{1}/\mathrm{51})-ϵ.

Key words： Waring-Goldbach problem; circle method; exponential sum over primes in short intervals

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