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Conference in memory of A. A. Karatsuba on number theory and applications
January 31, 2014 12:30–12:55, Moscow, Steklov Mathematical Institute, Conference-Hall

The distribution of the values of Dirichlet characters over the sequence of shifted primes

Z. Kh. Rakhmonov
 Video records: Flash Video 89.2 Mb Flash Video 533.6 Mb MP4 89.2 Mb

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:
$$T(\chi_q )=\sum_{p \leqslant x}\chi_q(p-l)\ll x\exp(-\sqrt{\mathcal{L}}).$$
This estimate improves the result of J.B. Friedlander, K. Gong and I.E. Shparlinski (2010), which is non -trivial only for $x\geqslant q^{ 8/9+\varepsilon}$.