Telescoping, rational-valued series, and zeta functions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Telescoping, rational-valued series, and zeta functions

Authors:	J. Marshall Ash Stefan Catoiu

Affiliation:	Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614 ; Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614

Abstract:	We give an effective procedure for determining whether or not a series $sum_{n=M}^{N}rleft( nright)$ telescopes when is a rational function with complex coefficients. We give new examples of series $left( astright) sum_{n=1}^{infty}rleft( nright)$ , where is a rational function with integer coefficients, that add up to a rational number. Generalizations of the Euler phi function and the Riemann zeta function are involved. We give an effective procedure for determining which numbers of the form are rational. This procedure is conditional on 3 conjectures, which are shown to be equivalent to conjectures involving the linear independence over the rationals of certain sets of real numbers. For example, one of the conjectures is shown to be equivalent to the well-known conjecture that the set is linearly independent, where $zetaleft( sright) =sum n^{-s}$ is the Riemann zeta function. Some series of the form $sum_{n}sleft( sqrt[r]{n},sqrt[r]{n+1} ,cdots,sqrt[r]{n+k}right)$ , where is a quotient of symmetric polynomials, are shown to be telescoping, as is . Quantum versions of these examples are also given.

Keywords:	Generalized zeta function generalized Euler phi function linear independence over the rationals

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