Telescoping, rational-valued series, and zeta functions

Abstract

We give an effective procedure for determining whether or not a series ∑ n = M N r ( n ) \sum _{n=M}^{N}r\left ( n\right ) telescopes when r ( n ) r\left ( n\right ) is a rational function with complex coefficients. We give new examples of series ( ∗ ) ∑ n = 1 ∞ r ( n ) \left ( \ast \right ) \sum _{n=1}^{\infty }r\left ( n\right ) , where r ( n ) 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 { ζ ( s ) : s = 2 , 3 , 4 , … } \left \{ \zeta \left ( s\right ) :s=2,3,4,\dots \right \} is linearly independent, where ζ ( s ) = ∑ n − s \zeta \left ( s\right ) =\sum n^{-s} is the Riemann zeta function. Some series of the form ∑ n s ( n r , n + 1 r , ⋯ , n + k r ) \sum _{n}s\left ( \sqrt [r]{n},\sqrt [r]{n+1} ,\cdots ,\sqrt [r]{n+k}\right ) , where s s is a quotient of symmetric polynomials, are shown to be telescoping, as is ∑ 1 / ( n ! + ( n − 1 ) ! ) \sum 1/(n!+\left ( n-1\right ) !) . Quantum versions of these examples are also given.