On the values of arithmetic functions in short intervals

Abstract. In this short paper the following assertion is proved. For positive integer \(d\) and \(c>0\) let \(J_c(N) = [N,~N+c\sqrt{N}]\) and \({\cal K}_d =\{n\in {\Bbb N}~\vert~(n,d)=1\}\). Let \(1 < N_1 < N_2 < \cdots\) be an infinite sequence of integers and \(\ell_1,\ell_2,\cdots\) be integers coprime to \(d\). Assume that \(f\) and \(g\) are completely additive functions defined on \({\cal K}_d\), for which \(f(n) = g(n)\) if \(n {\equiv} \ell_j\pmod d\), \(n\in J_c(N_j)\) \((j=1,2,\cdots )\). If \(c>2d\), then \(f(n)=g(n)\) identically on \({\cal K}_d\).