On the size of the Gelfond exponent

We use the explicit formula of V. Shevelev for the best possible exponent α(m)α(m) in the error term of the asymptotic formula of A.O. Gelfond on the number of positive integers n⩽xn⩽x in a given residue class modulo m and a given parity of the sum of its binary digits, to obtain new results about its behaviour. In particular, our result implies thatlim infp→∞α(p)=0 where p runs through the set of primes, which has been derived by V. Shevelev from Artin's conjecture.