Global maximum of ?(x)-x

[Joseph Biberstine]

Gerald McGarvey wrote:
> Based on a laborious semi-manual process (not recommended) of
> narrowing down the maximum, I believe the maximum of ?(x)-x
> occurs at a constant c whose continued fraction begins
> [0; 1, 3, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 
> 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 
> 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, ...
> I hope I didn't miss a larger ?(x)-x along the way.
> Here's a conjecture: the maximum occurs at a constant c whose continued 
> fraction
> starts as [0; 1, 3, 1, 4, 1, 4, 1, 5, then has the repeating sequence 1, 
> 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5
> so c = 1/(1+1/(3+1/(1+1/(4+1/(1+1/(4+1/(1+1/(5+1/((17325 + 
> 3*sqrt(59115595))/33461)))))))))
> or (2343*sqrt(59115595) + 18014599)/(2955*sqrt(59115595) + 22720034)
> or
> .79289414860601842644318261515336048837629147616654626681193944753721958311485553800434172526654160238532789219294579... 
> 
> If this is correct, the value of ?(c) appears to be 2008938429 / 
> 2147483647.
> Of course, it could very well be incorrect.
> 
> Has it be proven that ?(x) + ?(1-x) = 1 for all x?
> 
> - Gerry
<snip>
> 
Can't say why, but this doesn't feel right to me.  I would have expected 
a monotonic increase at even entries.  The repetition also seems 
artificial.  It is surely a good approximation, but I expected a 
non-quadratic irrational for c.