if ζ(1) is equal to 0, then there are an infinite number of rational points (solutions), and conversely, if ζ(1) is not equal to 0, then there is only a finite number of such points.

The statement you've provided seems to be related to the Riemann Zeta function, denoted as ζ(s). However, there seems to be a misunderstanding in your statement. The value of the Riemann Zeta function at 1, ζ(1), is actually a well-known problem called the Basel problem, and it is divergent, not equal to zero.

The Riemann Zeta function is defined as ζ(s) = 1^(-s) + 2^(-s) + 3^(-s) + ..., for complex number s with real part greater than 0. When s=1, this series does not converge, i.e., it tends towards infinity, not zero.

The connection between the Riemann Zeta function and the number of solutions to equations is a deep topic in number theory. The Riemann Hypothesis, one of the most famous unsolved problems in mathematics, conjectures that all non-trivial zeros of the Riemann Zeta function have real part 1/2. These zeros are connected to the distribution of prime numbers and many other deep results in number theory.

However, the value of ζ(1) does not directly tell us about the number of rational points or solutions to a particular equation. The number of solutions to an equation depends on the equation itself, not on the value of the Riemann Zeta function at a particular point.