# Solutions of a diophantine equation

zetafunction
given the diophantine polynomial equation

$$f(x)=0mod(p)$$

then is the number of solution approximately less than a given N approximately

$$\sum_{i\le N}e^{2i p\pi f(j)}$$

the idea is that the sum takes its maximum value every time p divides f(j) for some integer 'j''

Gold Member
I replied yesterday (March 26), but the reply was lost with the server problems. I'll try again:

Your summation expression appears to have a typo. If n is any integer, then

$$e^{2i p\pi n} = 1$$

so the expression always sums to N. Perhaps you meant to divide by p in the exponent instead of multiplying by p. Questions of this sort are discussed in the first few sections of Number Theory by Borevich and Shafarevich.

1. You might want to test your formula with $f(x) = x^p - x$, since all natural numbers are solutions.