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Selberg’s limit theorem for the Riemann zeta function on the critical line

The Riemann zeta function , defined for by and then continued meromorphically to other values of by analytic continuation, is a fundamentally important function in analytic number theory, as it is...

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254A, Notes 2: The central limit theorem

Consider the sum of iid real random variables of finite mean and variance for some . Then the sum has mean and variance , and so (by Chebyshev’s inequality) we expect to usually have size . To put it...

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A first draft of a non-technical article on universality

The month of April has been designated as Mathematics Awareness Month by the major American mathematics organisations (the AMS, ASA, MAA, and SIAM).  I was approached to write a popular mathematics...

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A second draft of a non-technical article on universality

I’ve spent the last week or so reworking the first draft of my universality article for Mathematics Awareness Month, in view of the useful comments and feedback received on that draft here on this...

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A central limit theorem for the determinant of a Wigner matrix

Van Vu and I have just uploaded to the arXiv our paper A central limit theorem for the determinant of a Wigner matrix, submitted to Adv. Math.. It studies the asymptotic distribution of the...

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275A, Notes 4: The central limit theorem

Let be iid copies of an absolutely integrable real scalar random variable , and form the partial sums . As we saw in the last set of notes, the law of large numbers ensures that the empirical averages...

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275A, Notes 5: Variants of the central limit theorem

In the previous set of notes we established the central limit theorem, which we formulate here as follows: Theorem 1 (Central limit theorem) Let be iid copies of a real random variable of mean and...

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