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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Telescoping, rational-valued series, and zeta functions


Authors: J. Marshall Ash and Stefan Catoiu
Journal: Trans. Amer. Math. Soc. 357 (2005), 3339-3358
MSC (2000): Primary 11J72, 11M41, 11A25, 40A25
Published electronically: March 10, 2005
MathSciNet review: 2135751
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Abstract: We give an effective procedure for determining whether or not a series $\sum_{n=M}^{N}r\left( n\right) $ telescopes when $r\left( n\right) $ is a rational function with complex coefficients. We give new examples of series $\left( \ast\right) \sum_{n=1}^{\infty}r\left( n\right) $, where $r\left( n\right) $ 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 $\left( \ast\right) $ 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 $\left\{ \zeta\left( s\right) :s=2,3,4,\dots\right\} $ is linearly independent, where $\zeta\left( s\right) =\sum n^{-s}$ is the Riemann zeta function.

Some series of the form $\sum_{n}s\left( \sqrt[r]{n},\sqrt[r]{n+1} ,\cdots,\sqrt[r]{n+k}\right) $, where $s$ is a quotient of symmetric polynomials, are shown to be telescoping, as is $\sum1/(n!+\left( n-1\right) !)$. Quantum versions of these examples are also given.


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Additional Information

J. Marshall Ash
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
Email: mash@math.depaul.edu

Stefan Catoiu
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
Email: scatoiu@math.depaul.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03699-8
PII: S 0002-9947(05)03699-8
Keywords: Generalized zeta function, generalized Euler phi function, linear independence over the rationals
Received by editor(s): August 12, 2003
Received by editor(s) in revised form: February 21, 2004
Published electronically: March 10, 2005
Additional Notes: The first author’s research was partially supported by NSF grant DMS 9707011 and a grant from the Faculty and Development Program of the College of Liberal Arts and Sciences, DePaul University.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.