On the size of the Gelfond exponent

We use the explicit formula of V. Shevelev for the best possible exponent α (m) in the error term of the asymptotic formula of A.O. Gelfond on the number of positive integers n ≤ 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 thatunder(lim inf, p → ∞) α (p) = 0 where p runs through the set of primes, which has been derived by V. Shevelev from Artin's conjecture.