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VIDEO LIBRARY |
À.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications
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Short exponential sums over primes Z. Kh. Rakhmonov Institute of Mathematics, Academy of Sciences of Republic of Tajikistan, Dushanbe |
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Abstract: I.M. Vinogradov was the first who obtained non-trivial estimates of the sum of values of non-principal character over the sequence of shifted primes. He proved that if $$ T(\chi)\,=\,\sum_{p\leqslant x}\chi(p-l)\,\ll\,x^{1+\varepsilon}\left(\sqrt{\frac{1}{q}+\frac{q}{x}}\,+\,x^{-1/6}\right). $$ This estimate is non-trivial for $$ T_{1}(\chi)\,\ll\,xq^{-\,\varepsilon^{2}/1024}. $$ In 2013, the author obtained non-trivial estimate of Theorem. Let $$ T(\chi)\,=\,\sum_{n\leqslant x}\Lambda(n)\chi(n-l)\,\ll\,x\exp{\bigl(-0.6\sqrt{\ln{D}}\bigr)}, $$ for |