For given positive integers n and a, let R(n;a) denote the number of
positive integer solutions (x,y) of the Diophantine equation naβ=x1β+y1β. Write S(N;a)=nβ€N(n,a)=1βββR(n;a). Recently Jingjing Huang and R. C. Vaughan proved that
for 4β€N and aβ€2N, there is an asymptotic formula S(N;a)=Ο2a3βpβ£aββp+1pβ1ββ N(log2N+c1β(a)logN+c0β(a))+Ξ(N;a). In this paper, we shall get a more explicit
expression with better error term for c0β(a)
In this paper we shall study the inverse problem relative to dynamics of the
w function which is a special arithmetic function and shall get some results.Comment: 11 page