Re: Difference equations, calculus, prime numbers

James Harris wrote:

<snip>


> In the limit as dy approaches 0, I have
>
> S'_y(x,y)= [p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y))
>
> and looking at
>
> p(x, y) = x - S(x, y) + C, 
>
> I can differentiate with respect to y to get
>
> p'_y(x,y) = - S'_y(x,y), 
>
> and making the substitution gives
>
> p'_y(x,y)= -[p(x/y, y) - p(y, sqrt(y))] p'_x(y, sqrt(y)),
>
> which is the partial differential equation.

Are you concerned by the fact that p has rather a lot of points
of discontinuity? Perhaps you should be. Derivatives of
discontinuous functions can be (ahem) somewhat ill-behaved,
as I'm sure you're aware.

<snip>


Regards,

Rick