TI-30XA calculator confirms proof of Riemann Hypothesis based on ZetaGrid data
We begin with Riemann's observation that the zeros of the zeta(s) function are equivalent to the zeros of the xi(t)
function, with s = 1/2 + ti. To prove that all zeros of the xi(t) function are real, therefore, is to prove the Riemann
Hypothesis. In fact, Riemann originally formulated his hypothesis in these terms, that all zeros of xi(t) are real.
Let us express t in terms of polar coordinates (r, theta): t = r cos(theta) + r sin(theta) i.
We know that -1/2 < r sin(theta) < 1/2 for all zeros of xi(t).
The data from the ZetaGrid distributed computing project show that all zeros of xi(t) are real for t < 2.4*10^12, and
therefore for r < 2.4*10^12.
Thus for any possible non-real zero of xi(t) with Re(t) and Im(t) positive (we may take this case without loss of
generality, for the zeros of xi(t) are symmetric about both axes of the complex plane), we know that
r sin(theta) < 1/2 and r > 2.4*10^12
sin(theta) < 1/(4.8*10^12)
sin(theta) < 2.083*10^-13.
Here is the key step of the proof, verified by the TI-30XA calculator:
For such small values of sin(theta), sin(theta) = theta. (We measure theta in radians, of course, not degrees.)
In fact, the TI-30XA confirms that even up to values as large as 1.4*10^-3, sin(theta) = theta holds. And here we are
dealing only with values of theta many orders of magnitude smaller than that.
Therefore, theta < 2.083*10^-13.
Further, the TI-30XA confirms that for such small values of theta, cos(theta) = 1. In fact this holds for values of theta
as large as 3.1*10^-5.
Therefore, for any possible non-real zero of xi(t) with Re(t) positive, sin(theta) = theta and cos(theta) = 1.
And for positive real zeros of xi(t) also, sin(theta) = theta and cos(theta) = 1.
Therefore, we may state without loss of generality that for all zeros of xi(t) with Re(t) positive, sin(theta) = theta
and cos(theta) = 1.
We now employ the trigonometric identity sin^2(theta) + cos^2(theta) = 1.
Thus we have for all zeros of xi(t) with Re(t) positive:
(theta)^2 + 1^2 = 1
(theta)^2 = 0
theta = 0
sin(theta) = 0.
Therefore for t = r cos(theta) + r sin(theta) i, the imaginary part is 0 for all zeros of xi(t) with Re(t) positive.
Thus all zeros of xi(t) with Re(t) positive are real.
Therefore, by the known symmetric property of the zeros of xi(t) about the imaginary axis, all zeros of xi(t) are real.
QED