. , Tang , X. H. and Zhang , J. , Existence of infinitely many solutions for elliptic boundary value problems with sign-changing potential , Electron. J. Differential Equations , 53 ( 2014 ), 1 – 11 .
It is known that under the assumption of the generalized Riemann hypothesis the function π(x,q,1) - π(x,q,a) has infinitely many sign changes. In this article we give an upper bound for the least such sign change. Similarly, assuming
the Riemann hypothesis we give a lower bound for the number of sign changes of π(x)-li x. The implied results for the least sign change are weaker than those obtained by numerical methods, however, our method makes
no use of computations of zeros of the ζ-function.
The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in
Lpspaces for 1 < p < ∞ and proves that if f(x)∈Lp[-1,1] changes sign exactly l times in (-1,1), then there exists r(x)∈Rnl such that
where Rnl indicates all rational functions whose denominators consist of polynomials of degree n and numerators polynomials of degree l, and Cp, l,b is a positive constant depending only on p, l and b which relates to the distance among the sign change points of f(x) and will be given in 3.