Geodesic
Short Sums with a Noninteger Power of a Natural Number
Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 763-774

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We establish a nontrivial estimate for a short trigonometric sum of the form $\sum_{x-y$, where $y\ge \sqrt{2cx}\,{\mathscr L}^A$, $A\ge 1$ is a fixed number, ${\mathscr L}=\ln x$, and $c$ is a noninteger satisfying the conditions $$ 1\le \log_2{\mathscr L}-\log_2 \ln {\mathscr L}^{6A},\qquad \|c\|\ge(2^{[c]+1}-1)(A+1){\mathscr L}^{-1}\ln{\mathscr L}. $$
Keywords: short trigonometric sum, estimate of a trigonometric sum, Fourier series, Stirling's formula. 
@article{MZM_2014_95_5_a10,
     author = {P. Z. Rakhmonov},
     title = {Short {Sums} with a {Noninteger} {Power} of a {Natural} {Number}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {763--774},
     publisher = {mathdoc},
     volume = {95},
     number = {5},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a10/}
}
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P. Z. Rakhmonov. Short Sums with a Noninteger Power of a Natural Number. Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 763-774. http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a10/