Video library: P. Z. Rakhmonov, Short exponential sums with a fractional powers of natural numbers

VIDEO LIBRARY


Conference in memory of A. A. Karatsuba on number theory and applications January 31, 2014 16:00, Moscow, Steklov Mathematical Institute, Lecture Room 530

Short exponential sums with a fractional powers of natural numbers P. Z. Rakhmonov
Abstract: The talk is devoted to the estimation of the exponential sum of the type $$ \sum_{x-y<n\le x}\exp(2\pi i\alpha [n^c]), $$ where $y\ge\sqrt{2cx}\,\mathcal{L}^{A}$, $A\ge 1$ is fixed, $\mathcal{L}=\ln x$ and $c$ is non-integer satisfying the conditions $$ 1<c\leqslant \log_2\mathcal{L}-\log_2\ln(\mathcal{L}^{6A}) , \qquad \\|c\\|\ge (2^{[c]+1}-1)(A+1) \mathcal{L}^{-1}\ln{\mathcal{L}}. $$