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. , 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 . [5

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Abstract  

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.

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11 35 47 Lü, H., O’Regan, D. and Agarwal, R. P. , A positive solution for singular discrete boundary value problems with sign changing

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-Poisson systems , J. Differential Equations , 248 ( 2010 ), 521 – 543 . [9] Chen , S. T. and Tang , X. H. , Ground state sign-changing

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Abstract  

The present paper establishes a complete result on approximation by rational functions with prescribed numerator degree in L pspaces for 1 < p < ∞ and proves that if f(x)∈L p [-1,1] changes sign exactly l times in (-1,1), then there exists r(x)∈R n l such that

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\| {f(x) - r(x)} \right\|_{L^p } \leqq C_{p,l,b} \omega (f,n^{ - 1} )_{L^p } ,$$ \end{document}
where R n l indicates all rational functions whose denominators consist of polynomials of degree n and numerators polynomials of degree l, and C p , 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.

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] Chung , N. T. Ngo , Q. A. 2009 A multiplicity result for a class of equations of p -Laplacian type with sign-changing nonlinearities Glasgow Math. J

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Yang, J. P. , Sign-changing solutions to discrete fourth-order Neumann boundary value problems, Adv. Difference Equ. , 2013 (2013), 1–10. J. P Y. Sign-changing solutions to

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