Beatty Log Series

[Paul D. Hanna]

Seqfans,
      Can someone transform the following series: 
   Sum_{n>=1} ( 1/[n*x] - 1/(n*x) ) 
into one that converges more rapidly than 
   Sum_{n>=1} FractionalPart(n*x) / (n*x*[n*x]) 
for irrational x >1 ?
 
The series involves reciprocal terms of a Beatty sequence
and has the following interesting property.

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Given an irrational number t, 0<t<1, 
define a function B(t) by the series 
   B(t) = Sum_{n>=1} ( 1/[n(1+t)] - 1/(n(1+t)) )
then 
   B(t) + B(1/t) = log(1+t) - log(t)*t/(1+t) 
where [x] denotes the integer floor of x. 
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I do not know what B(t) is in itself, 
but it can be expressed as: 
   B(t) = log(1+t)/(1+t) - L(t) 
   B(1/t) = log(1+1/t)/(1+1/t) - L(1/t) 
where L is an unknown function that satisfies 
   L(1/t) = - L(t).
Now I wonder, what is the function L(t)?
 
Note: the sum of differences of reciprocal Beatty sequences
  B(t) - B(1/t) = Sum_{n>=1} ( 1/[n(1+t)] - 1/[n(1+1/t)] )
   =  log(1+t)*(1-t)/(1+t) + log(t)*t/(1+t) - 2*L(t)
may be a clue to the mysterious L(t).
   
Perhaps there are 'constants' involved here 
analogous to the Euler gamma constant ... 
 
Taking the analogy even further, 
wonder if a "Beatty Zeta function": 
   Z(s,t) = Sum_{n>=1} 1/[n*(1+t)]^s 
where
   Z(s,t) + Z(s,1/t) = zeta(s) 
has been studied before (references?). 
  
Any comments would be appreciated.
Thanks,
     Paul