Résumés

"Shifted convolutions and exponential sums"
Sary Drappeau
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The shifted convolution problem asks for an asymptotic formula for the average $$\sum_{n\leq x} f(n) g(n+1) $$ of the shifted product of two multiplicative functions. In these courses we will cover some cases of this problem, focusing on the case $g(n) = \tau(n) = 1* 1(n)$, the number of divisors of n. This will mainly be an alibi to talk about equidistribution in arithmetic progressions, and algebraic exponential sums, in particular (but not only) Kloosterman sums.

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"Probabilistic aspects of character sums"
  Adam Harper
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Sums of Dirichlet characters are amongst the most studied quantities in analytic number theory. Their maximum possible size is connected with the classical Polya--Vinogradov inequality, the Burgess bound, and the Generalised Riemann Hypothesis. There has also been great interest in the "distribution" of character sums in various senses, for example where the character varies, or the interval of summation moves, or both. All of these problems require a good understanding of which random objects properly model sums of Dirichlet characters in different settings, and how those random models behave. In my lectures I will try to explain both classical and modern ideas on these various problems, including the "better than squareroot cancellation" for character sums that flows from their connection with critical multiplicative chaos. I will also describe some open problems.

Probabilistic aspects character sums talk 1 Slides

Probabilistic aspects character sums talk 2 Slides

Probabilistic aspects character sums talk 3 Slides

Probabilistic aspects character sums talk 4 Slides

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"The method of Type I/II sums and sieves"
 James Maynard
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The only reasonably general method we have for studying primes in a set of integers is the method of Type I/II sums, first pioneered by Vinogradov but then developed by many others. In its modern form, this method asks us to first calculate some 'arithmetic information' about the set in question, and then to try to use sieve ideas to convert this arithmetic information into estimates for the number of primes in the original set.

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Talk 4

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"L-functions, sieves and branching random walks"
Maksym Radziwill
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I will discuss the recent (and older) connections between sieve
theory and the behavior of L-functions. I will illustrate how this connection
can be used to prove concrete results by explaining some parts of my
recent work with Arguin and Bourgade on the Fyodorov-Hiary-Keating conjecture
about the local maximum of the Riemann zeta-function on the critical line. In turn
the Fyodorov-Hiary-Keating conjecture is related to branching random walks and
our resolution of the Fyodorov-Hiary-Keating conjecture sheds new light on branching
phenomena even in the purely probabilistic setting of random matrices. This shows
how a connection between two purely number theoretic topics can percolate into an
advance in analysis.

Radziwill slide1

Radziwill slide2

Radziwill slide3

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 "Maxima of zeta in short-ish intervals: numerics and large-deviation corrections"
  Emma Bailey (Bristol University, UK)

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 The Fyodorov-Hiary-Keating conjecture on the maximum of the Riemann zeta function over short (i.e. O(1)) ranges has seen much progress over the past decade.  A generalisation to the conjecture, extending to ranges of logarithmic sizes, was proposed by Arguin-Dubach-Hartung based on the analysis of a related probabilistic model.  We present numerics in support of this generalised conjecture, and suggest a further refinement due to large deviations to Selberg's CLT (following Radziwiłł).  Such corrections are also present in the associated random matrix model.  This work is joint with Amzallag-Arguin-Hui-Rao.

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"A weighted one-level density of the non-trivial zeros of $L$-functions"
Alessandro Fazzari (Genova University, Italy)

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We consider a weighted version of the one-level density of the non-trivial zeros of $L$-functions, tilted by a power of the $L$-function evaluated at the central point. More precisely, for three specific families of $L$-functions (order by log-conductor $c(L)$) with different symmetry types, assuming the Riemann Hypothesis and the ratio conjecture for these families, we investigate the quantity $$\frac{1}{\sum\limits_{L\in\mathcal F}L(\tfrac{1}{2})^k}\sum_{L\in\mathcal F}\sum_{\gamma_L}f(c(L)\gamma_L)L(\tfrac{1}{2})^k $$ with $k$ a positive integer, $f$ a test function and $\gamma_L$ the imaginary part of a generic non-trivial zero of $L$.

Note

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"An almost sure upper bound for random multiplicative functions on integers with a large prime factor"
Daniele Mastrostefano (Warwick University, UK)

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Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.

Talk

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 "Distribution of primes in thin subsets and exponential sums"
 Andrei Shubin (Caltech University, USA)

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 In the talk we will discuss some questions related to exponential sums over primes and the distribution

 of primes with the restriction on the fractional part of $p^{\alpha}$ for non-integer $\alpha > 0$.

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