NTIC Epilogue

Section 25.8 Epilogue

The Riemann zeta function and counting primes is truly only the beginning of research in modern number theory. Let's see just a little more of its future.

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For instance, research in finding and counting points on curves (as in Chapter 15) leads to more complicated series like

ζ,

called

L

-functions. There is a version of the Riemann Hypothesis for them, too (see Fact 25.7.3 for some connections). Even without that, they gives truly interesting, strange, and beautiful results, particularly when counting points on the elliptic curves we mentioned at various points in text; a notable success of this was in the proof of Fermat's Last Theorem. You may wish to continue with books like [E.4.19] or [E.4.5, Section 12.4] or [E.4.24, Chapters 13-15], or perhaps start doing Exercise 25.9.9 with an internet search.

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Here is a recent result of interest. Recall from Example 14.2.3 that the notation

r_{12} (n)

should denote the number of ways to write

n

as a sum of twelve squares. Here, order and sign both matter, so

(1, 2)

and

(2, 1)

and

(- 2, 1)

are all different.

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Theorem 25.8.1.

As we let $p$ run through the set of all prime numbers, the distribution of the fraction

\frac{r_{12} (p) - 8 (p^{5} + 1)}{32 p^{5 / 2}}

is precisely as this circular function in the long run:

\frac{2}{π} \sqrt{1 - t^{2}}

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Proof.

Needless to say, this result is far beyond the level of this text – but maybe you will make the next contribution? Initially this result is a corollary of the proof of the Sato-Tate conjecture by Barnet-Lamb, Geraghty, Harris, and Taylor; that proof crucially used the so-called “Fundamental Lemma” of Gérard Laumon and Ngô Bảo Châu, the latter of whom won the Fields Medal based on proving it in very full generality.

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Sage note 25.8.2. Into the future.

The following graphic is based on one due to William Stein, the original founder and developer of Sage, in personal communication.

Figure 25.8.3. Distribution of modified twelve squares fraction (Sato-Tate)

Try it interactively below. The higher the number, the closer the values should group to the distribution; change the number of bins in the histogram to see it more clearly.

xxxxxxxxxx
 
def sqrt2():
    PI = float(pi)
    return plot(lambda x: (2/PI)*math.sqrt(1-x^2), -1,1, plot_points=200,
              rgbcolor=(0.3,0.1,0.1), thickness=2)
​
delta = delta_qexp(10^5)
​
@interact
def delta_dist(bins=(20,[10..150]), number = [500,1000,..,delta.prec()]):
    D = delta[:number]
    w = [float(D[p])/(2*float(p)^(5.5)) for p in prime_range(number + 1)]
    show(histogram(w, bins=bins, density=True) + sqrt2(), frame=True, gridlines=True)

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What an amazing result. These ideas are at the forefront of all types of number theory research today, and my hope is that you will enjoy exploring more of it, both with paper and pencil and using tools like Sage!