 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 357, Number 8, Pages 33393358
S 00029947(05)036998
Article electronically published on March 10, 2005
TELESCOPING, RATIONALVALUED SERIES,
AND ZETA FUNCTIONS
J. MARSHALL ASH AND STEFAN CATOIU
Abstract. We give an effective procedure for determining whether or not a
series N
n=M r (n) telescopes when r (n) is a rational function with complex
coefficients. We give new examples of series ()
n=1 r (n), where r (n) 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 wellknown conjecture that the
