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## Hardy and the zeta function, Part 2

After that brief hiatus, we return to the proof of Hardy’s theorem that the Riemann zeta function has infinitely many zeros on the real line; probably best to go and brush up on part one first. Two things remain to … Continue reading

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## Symmetrization

We call a random variable (which is just a measurable function where is a probability measure space) symmetric if is identically distributed to . Symmetric random variables are often a lot more pleasant to handle. Much of what is true … Continue reading

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## Every large number is the sum of 13 cubes

An important function that arises from Waring’s problem is the the following. For any , let be the least such that, for all sufficiently large (how large can depend on and ) there exist positive integers such that Waring’s problem … Continue reading

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## Hardy and the zeta function, Part 1

Today we’ll begin with an interesting and important theorem about the Riemann zeta function. Recall that this is defined when by the absolutely convergent series This can be analytically continued to a function meromorphic on the entire complex plane, holomorphic … Continue reading

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## Szemerèdi Regularity Lemma

Welcome to the first in our series about Great Lemmas of Mathematics. We’ll start with one of the greats from combinatorics, perhaps the most useful tool in combinatorics and graph theory: the Szemerèdi Regularity Lemma (SRL). This was first proved … Continue reading

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## Rudin’s inequality

After that whirlwind tour through Khintchine’s inequality, let’s take a look at the less well-known Rudin’s inequality. This says (at least in one form) that, given a finite abelian group and a function if the support of the Fourier transform … Continue reading

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## Khintchine’s inequality

It was recently suggested to me that the proof of Khintchine’s inequality is one which I should think about deeply, and I would especially like to explore the relationship between Khintchine’s and Rudin’s inequality. This will be the first of … Continue reading

## What does additive structure mean?

Suppose we have a finite subset of an abelian group, and I told you it was additively structured. There are lots of different things this vague term could mean: The set is not much bigger than . There are lots … Continue reading