Monthly Archives: January 2016

Math 420: First Week Homework and References

Firstly, there are three administrative notes.

  1. I’ve posted the first homework set. This is due on Thursday, and you can find it here.
  2. I haven’t set official office hour times yet. But I will have office hours on Monday from noon to 2pm on Monday, 1 Feb 2016, in my office at the Science Library.
  3. If you haven’t yet, I encourage you to read the syllabus.

We mentioned several good and interesting “number theoretic” problems in class today. I’d like to remind you of some of them, and link you to some additional places for information.

Pythagorean Theorem

We’ve found all primitive Pythagorean triples in integers, which is a very nice theorem for an hour. But I also mentioned some of the history of the Pythagorean Theorem and the significance of numbers and number theory to the Greeks.

I told the class a story about how the Pythagorean student who revealed that there were irrational numbers was stoned. This is apocryphal. In fact, there is little exact record, but his name was Hippasus and it is more likely that he was drowned for releasing this information.

For this and other reasons, the Pythagorean school of thought split into two sects, one from Pythagoras and one from Hippasus.

Goldbach’s Conjecture

Is it the case that every even integer is the sum of two primes? We think so. But we do not know.

I mentioned the Ternary Goldbach Conjecture, also known as the Weak Goldbach Conjecture, which says that every odd integer greater than $5$ is the sum of three odd primes. This was proved very recently. If you’re interested in what a mathematical paper looks like, you can give this paper a look. [Do not expect to be able to understand the paper — but it is interesting what sorts of tools can be used towards number theory]

Fermat’s Last Theorem

Are there nontrivial integer solutions to $X^n + Y^n = Z^n$ where $n \geq 3$?

This is one of the most storied and studied problems in mathematics. I think this has to do with how simple the statement looks. Further, we managed to fully classify all solutions when $n = 2$ in one class period. It doesn’t seem like it should be too hard to extend that to other exponents, does it?

If time and interest permits, we will return to this topic at the end of the course. There is no way that we could present a proof, or even fully motivate the proof. But we might be able to say a few words about how progress towards the theorem spurred and created mathematics, and maybe we can give a hint of the breadth of the ideas used to finally produce a proof.

Twin Prime Conjecture

Are there infinitely many primes $p$ such that $p+2$ is also prime? We think so, but we don’t know. Two years ago, we had absolutely no idea at all. Then Yitang Zhang had a brilliant idea (and not much later a graduate student named James Maynard had another brilliant idea) which allowed some sort of progress.

This culminated with the Polymath8 Project Bounded Gaps Between Primes. Math can be a social sport, and the polymath projects are massively collaborative online and open projects towards math problems. They’re still a bit new, and a bit experimental. But Polymath8 is certainly extremely successful.

What is known is that there exists at least one even number $H \leq 246$ such that $p$ and $p + H$ is prime infinitely often. In fact, James Maynard showed that you can make more complicated ensembles of prime distances.

The ideas that led to this result can likely be sharpened to give better results, but actually proving that there are infinitely many twin primes is almost certainly going to require a brand new idea and methodology.

The best related result comes from Chinese mathematician Chen Jingrun, who proved that every sufficiently large even integer can be written either as a sum of two primes, or as a sum of a prime and a number with exactly two prime factors. Although this seems very close, it is also likely that this idea cannot be sharpened further.

Writing Numbers as Sums of Squares, Cubes, and So On

Can every integer be written as the sum of three squares? What about four squares? More generally, is there a number $n$ so that every integer can be written as a sum of at most $n$ squares?

Similarly, is there a number $n$ so that every integer can be written as a sum of at most $n$ cubes? What about fourth powers?

These problems are all associated to something called Waring’s Problem, about which much is known and much is unknown.

We also asked which primes can be written as a sum of two squares. Although we might have a hard time finding those primes right now, you might try to show that if $p$ is a prime that can be written as a sum of two squares, then either $p$ is $2$, or $p = 4z + 1$ for some integer $z$. The reasoning is very similar to some of the reasoning done in class today.

Max’s Conjecture

For primitive Pythagorean triples $(a,b,c)$ with $a^2 + b^2 = c^2$, we showed that we can restrict out attention to cases where $a$ is odd, $b$ is even, and $c$ is odd. Max conjectured that those $c$ on the right are always of the form $4k + 1$ for some $k$, or equivalently $c$ is always an integer that leaves remainder $1$ after being divided by $4$.

We didn’t return to this in class, but we can now. First, note that since $c$ is odd, we can write $c$ as $2z + 1$ for some $z$. But we can do more. We can actually write $c$ as either $4z + 1$ or $4z + 3$. (Can you prove this?)

Max conjectured that it is always the case that $c = 4z + 1$. So we might ask, “What if $c = 4z + 3$?”

Writing $a = 2x + 1$ and $b = 2y$, we get the equation

$$ \begin{align}
a^2 + b^2 &= c^2 \\
4x^2 + 4x + 1 + 4y^2 &= 16z^2 + 24z + 9, \end{align}$$

which can be rewritten as
$$ 4x^2 + 4x + 4y^2 = 16z^2 + 24z + 2.$$
You can divide by $2$. Then we ask: what’s the problem? Why is this bad? (It is, and it’s very similar to some questions we asked in class.)

So Max’s Conjecture is true, and every number appearing as $c$ in a primitive Pythagorean triple is of the form $c = 4z + 1$ for some integer $z$.

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Paper:Short-Interval Averages of Sums of Fourier Coefficients of Cusp Forms

This is joint work with Thomas Hulse, Chan Ieong Kuan, and Alex Walker, and is a sequel to our previous paper.

We have just uploaded a paper to the arXiv on estimating the average size of sums of Fourier coefficients of cusp forms over short intervals. (And by “just” I mean before the holidays). This is the second in a trio of papers that we will be uploading and submitting in the near future.

Suppose $latex {f(z)}$ is a weight $latex {k}$ holomorphic cusp form on $\text{GL}_2$ with Fourier expansion

$$f(z) = \sum_{n \geq 1} a(n) e(nz).$$

Denote the sum of the first $n$ coefficients of $f$ by $$S_f(n) := \sum_{m \leq n} a(m). \tag{1}$$
We consider upper bounds for the second moment of $latex {S_f(n)}$ over short intervals.

In our earlier work, we mentioned the conjectured bound $$ S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}, \tag{2}$$
which we call the “Classical Conjecture.” There has been some minor progress towards the classical conjecture in recent years, but ignoring subpolynomial bounds the best known result is of shape $$ S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}}. \tag{3}$$

One can also consider how $latex {S_f(n)}$ behaves on average. Chandrasekharan and Narasimhan [CN] proved that the Classical Conjecture is true on average by showing that $$ \sum_{n \leq X} \lvert S_f(n) \rvert^2 = CX^{k- 1 + \frac{3}{2}} + B(X), \tag{4}$$
where $latex {B(x)}$ is an error term. Later, Jutila [Ju] improved this result to show that the Classical Conjecture is true on average over short intervals of length $latex {X^{\frac{3}{4} + \epsilon}}$ around $latex {X}$ by showing $$ X^{-(\frac{3}{4} + \epsilon)}\sum_{\lvert n – X \rvert < X^{3/4} + \epsilon} \lvert S_f(n) \rvert^2 \ll X^{\frac{k-1}{2} + \frac{1}{4}}. \tag{5}$$
In fact, Jutila proved a much more complicated set of bounds, but this bound can be read off from his work.

In our previous paper, we introduced the Dirichlet series $$ D(s, S_f \times S_f) := \sum_{n \geq 1} \frac{S_f(n) \overline{S_f(n)}}{n^{s + k – 1}} \tag{6}$$
and provided its meromorphic continuation In this paper, we use the analytic properties of $latex {D(s, S_f \times S_f)}$ to prove a short-intervals result that improves upon the results of Jutila and Chandrasekharan and Narasimhan. In short, we show the Classical Conjecture holds on average over short intervals of width $latex {X^{\frac{2}{3}} (\log X)^{\frac{2}{3}}}$. More formally, we prove the following.

Theorem 1 Suppose either that $latex {f}$ is a Hecke eigenform or that $latex {f}$ has real coefficients. Then \begin{equation*} \frac{1}{X^{\frac{2}{3}} (\log X)^{\frac{2}{3}}} \sum_{\lvert n – X \rvert < X^{\frac{2}{3}} (\log X)^{\frac{2}{3}}} \lvert S_f(n) \rvert^2 \ll X^{\frac{k-1}{2} + \frac{1}{4}}. \end{equation*}

We actually prove an ever so slightly stronger statement. Suppose $latex {y}$ is the solution to $latex {y (\log y)^2 = X}$. Then we prove that the Classical Conjecture holds on average over intervals of width $latex {X/y}$ around $latex {X}$.

We also demonstrate improved bounds for short-interval estimates of width as low as $latex {X^\frac{1}{2}}$.

There are two major obstructions to improving our result. Firstly, we morally use the convexity result in the $latex {t}$-aspect for the size of $latex {L(\frac{1}{2} + it, f\times f)}$. If we insert the bound from the Lindel\”{o}f Hypothesis into our methodology, the corresponding bounds are consistent with the Classical Conjecture.

Secondly, we struggle with bounds for the spectral component $$ \sum_j \rho_j(1) \langle \lvert f \rvert^2 y^k, \mu_j \rangle \frac{\Gamma(s – \frac{3}{2} – it_j) \Gamma(s – \frac{3}{2} + it_j)}{\Gamma(s-1) \Gamma(s + k – 1)} L(s – \frac{3}{2}, \mu_j) V(X, s) \tag{7}$$
where $latex {\mu_j}$ are a basis of Maass forms and $latex {V(X,s)}$ is a term of rapid decay. For our analysis, we end up bounding by absolute values and are unable to understand cancellation from spin. An argument successfully capturing some sort of stationary phase could significantly improve our bound.

Supposing these two obstructions were handled, the limit of our methodology would be to show the Classical Conjecture in short-intervals of width $latex {X^{\frac{1}{2}}}$ around $latex {X}$. This would lead to better bounds on individual $latex {S_f(X)}$ as well, but requires significant improvement.

For more details and specific references, see the paper on the arXiv.

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