Video library: Z. Kh. Rakhmonov, The distribution of the values of Dirichlet characters over the sequence of shifted primes

VIDEO LIBRARY


Conference in memory of A. A. Karatsuba on number theory and applications January 31, 2014 12:30, Moscow, Steklov Mathematical Institute, Conference-Hall

The distribution of the values of Dirichlet characters over the sequence of shifted primes Z. Kh. Rakhmonov
Abstract: The talk is devoted to the following result of the speaker: Theorem. Let $q$ be a sufficiently large natural number, and suppose that $\chi_{q}$ is a primitive character modulo $q$. Suppose also that $(l,q)=1$, and let $\varepsilon$ be arbitrary small positive constant, $\mathcal{L}\,=\,\ln q$, $x\geqslant q^{\,5/6+\varepsilon}$. Then we have: $$ T(\chi_q )=\sum_{p\,\leqslant\, x}\chi_q(p-l)\ll x\exp\bigl(-\sqrt{\mathcal{L}}\bigr). $$ This estimate improves the result of J.B. Friedlander, K. Gong and I.E. Shparlinski (2010), which is non -trivial only for $x\geqslant q^{\,8/9+\varepsilon}$.