06/30/2011, 12:27 PM
to test this it seems the (q-)continuum sum defined by mike ( rediscovered ) is not practical in a numerical sense.
however there are other continuum sum methods that give the same result as long as the values are real and twice differentiable.
those others give the same numerical value ( proof ref desired ) as for instance " Riemann's continuum sum " :
sum a,b f(x) = integral a,b f(x) dx + integral f ' (x) ( x - floor(x) -1/2) dx + (f(a) + f(b))/2
if 'a' is an integer and f ' (x) is continu.
together with log^[3/2] = around log log log log 2sinh^(1/2) ( exp exp x ) , it seems the OP conjecture can be numericly tested !
sorry for no tex use , im in a hurry.
regards
tommy1729
however there are other continuum sum methods that give the same result as long as the values are real and twice differentiable.
those others give the same numerical value ( proof ref desired ) as for instance " Riemann's continuum sum " :
sum a,b f(x) = integral a,b f(x) dx + integral f ' (x) ( x - floor(x) -1/2) dx + (f(a) + f(b))/2
if 'a' is an integer and f ' (x) is continu.
together with log^[3/2] = around log log log log 2sinh^(1/2) ( exp exp x ) , it seems the OP conjecture can be numericly tested !
sorry for no tex use , im in a hurry.
regards
tommy1729
