SUMS OF CHARACTERS MODULO A CUBEFREE AT SHIFTED PRIMES

Vinogradov’s method of estimation of exponential sums over primes allowed him to solve the number of arithmetic problems with primes. One of them is a problem of distribution of the values of non-principal character on the sequence of shifted primes. In 1938 he proved that if q is an odd prime, (l, q) = 1, χ(a) is non-principal character modulo q, then T(χ) = X p6x χ(p − l) ≪ x1+ε r 1 q + q x + x−1 6  . (IMV ) This estimate is non-trivial when x ≫ q1+ε and an asymptotic formula for the the number of quadratic residues (non-residues) modulo q of the form p − l, p 6 x follows from it. Later in 1953, I. M. Vinogradov obtained a non-trivial estimate of T(χ) when x > q0,75+ε, q is a prime. It was a surprising result. In fact, T(χ) can be represented as a sum over zeroes of correspondent Dirichlet L — function; So a non-trivial estimate of T(χ) is obtained only for x > q1+ε provided that the extended Riemann hypothesis is true. In 1968 A. A. Karatsuba found a method that allowed him to obtain non-trivial estimate of short sums of characters in finite fields with fixed degree. In 1970 using the modification of his technique coupled with Vinogradov’s method he proved that: if q is a prime number, χ is non-principal character modulo q and x > q 1 2+ε, then the following estimate is true T(χ) ≪ xq− 1 1024 ε2 . In 1985 Z. Kh. Rakhmonov generalized the estimate (IMV) for the case of composite modulo and proved: let D is a sufficiently large positive integer, χ is a non-principal character modulo D, χq is primitive character generated by character χ, then
T(χ) 6 x ln5 x r 1 q + q x τ 2(q1) + x−1 6 τ (q1)  , q1 = p\D p ̸ \q 
p. If a character χ coincides with it generating primitive character  q, then the last estimate is non-trivial for x > q(ln q)13. In 2010 г. J. B. Friedlander, K. Gong, I. E. Shparlinski showed that a non-trivial estimateof the sum T(χq) exists for composite q when x — length of the sum, is of smaller order than q. They proved: for a primitive character χq and an arbitrary ε > 0 there exists such δ > 0 that for all x > q 8 9+ε the following estimate holds: T(χq) ≪ xq−δ. In 2013 Z. Kh. Rakhmonov obtained a non-trivial estimate of T(χq) for the composite modulo q and primitive character χq when x > q 5 6+ε. In this paper the theorem about the estimate of the sum  T(χq) is proved for cubefree modulo q. It is non-trivial when x > q 5 6+ε.

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