This is the webpage for the grad seminar of the focused research group on averages of $$L$$-functions and arithmetic stratification funded by the NSF grant DMS-1854398, centered at the American Institute of Mathematics.

This an online seminar aimed at graduate students in the FRG, though non-grad students often attend. It is meant for blackboard-style expository talks; typically, past speakers have either given an overview of a general field or went over influential papers with a focus on techniques they find useful and wanted to advertise.

If you would like to volunteer to give a talk, or you would like to attend the seminar, please send me an email.

For the schedule as well as notes and recordings of past talks, please see below.

# Next Talk [06-Feb-2023]

## Keshav Aggarwal (Rényi Institute)

### “Bound for the existence of prime gap graphs”

Abstract: Given a sequence $$\mathbf{D}$$ of non-negative integers, it is interesting to know whether there exists a graph with vertices of degrees equaling the integers in $$\mathbf{D}$$. If that happens, we say $$\mathbf{D}$$ is graphic. Clearly, if the sequence is graphic, then the sum of its members must be even. However, it is not self-evident whether a given sequence is graphic. There are exponentially many different realizations for almost every graphic degree sequence. At the same time, the number of all graphic degree sequences is infinitesimal compared to the number of integer partitions of the sum of the degrees. Therefore it is incredibly hard to come up with an interesting (or non-trivial) graphic degree sequence.

Let us call a simple graph on $$n>2$$ vertices a prime gap graph if its vertex degrees are $$1$$ and the first $$n-1$$ prime gaps. Recently, Erdős-Harcos-Kharel-Maga-Mezei-Toroczkai showed that the prime gap sequence is graphic for large enough $$n$$. In a joint work with Robin Frot, we make their work effective.

# Upcoming Talks

## Spring 2023

“TBA”

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### [12-Jun-2023] Andrés Chirre (Rochester)

“TBA”

$$\star$$ – to be confirmed

# Past Talks

## Spring 2023

### [30-Jan-2023] David Farmer (AIM)

“The zeta function when it is particularly large” (notes, video)

### [23-Jan-2023] Nathan Ng (Lethbridge)

“The eighth moment of the Riemann zeta function” (notes, video)

## Fall 2022

### [19-Dec-2022] George Dickinson (Manchester)

“Second moments of Dirichlet L-functions” (notes, video)

### [12-Dec-2022] Alexander Dobner (Michigan)

“Optimization and moment methods in number theory” (notes, video)

### [05-Dec-2022] Asif Zaman (Toronto)

“Random multiplicative functions and a simplified model”

### [28-Nov-2022] Aled Walker (KCL)

“Correlations of sieve weights and distributions of zeros” (notes, video)

### [21-Nov-2022] Alexander Dunn (Caltech)

“Bias in cubic Gauss sums: Patterson's conjecture” (video)

### [14-Nov-2022] Jakob Streipel (Maine)

“Using second moments to count zeros” (notes, video)

### [07-Nov-2022] Daodao Yang (TU Graz)

“Large values of derivatives of the Riemann zeta function and related problems” (notes)

### [31-Oct-2022] Emma Bailey (CUNY)

“Large values of $$\zeta$$ on the critical line” (video)

### [17-Oct-2022] Dan Goldston (SJSU)

“Small gaps and spacings between Riemann zeta-function zeros” (notes, video)

### [10-Oct-2022] Lasse Grimmelt (Oxford)

“Primes in large arithmetic progressions and applications to additive problems” (notes, video)

### [03-Oct-2022] Louis Gaudet (Rutgers)

“The least Euler prime via a sieve approach” (video)

### [26-Sep-2022] Hua Lin (UC Irvine)

“One-level density of zeros of Dirichlet L-function over function fields” (notes, video)

### [22-Sep-2022] Caroline Turnage-Butterbaugh (Carleton)

“Moments of Dirichlet L-functions” (notes)

## Spring 2022

### [30-May-2022] Micah Milinovich (Mississippi)

“Estimates for zeta and L-functions via Fourier optimization” (notes, video)

### [23-May-2022] Jori Merikoski (Oxford)

“Primes in sparse polynomials sets” (notes, video)

### [16-May-2022] Martin Čech (Concordia)

“Two ways to compute moments in the family of real Dirichlet L-functions” (notes, video)

### [02-May-2022] Alexandra Florea (UC Irvine)

“Upper bounds for positive and negative moments of the Riemann zeta function” (notes, video)

### [25-Apr-2022] Kevin Hughes (Bristol)

“A primer on Discrete Restriction Theory” (notes, video)

### [18-Apr-2022] Alia Hamieh (UNBC)

“Distribution of values of logarithmic derivatives of L-functions” (notes, video)

### [11-Apr-2022] Ayla Gafni (Mississippi)

“Uniform distribution and geometric incidence theory” (video)

### [04-Apr-2022] Victor Wang (Princeton)

“Special subvarieties over finite and infinite fields” (notes, video)

### [28-Mar-2022] Joshua Stucky (Kansas State)

“Smooth numbers in arithmetic progressions” (notes, video)

### [21-Mar-2022] Kunjakanan Nath (UIUC)

“Primes with restricted digits” (notes, video)

### [14-Mar-2022] Ofir Gorodetsky (Oxford)

“Chebyshev's bias for primes and for sums of two squares” (notes, video)

### [07-Mar-2022] Vivian Kuperberg (Stanford)

“The Hardy–Littlewood $$k$$-tuple conjecture and intervals with many primes” (notes, video)

### [28-Feb-2022] Peter Humphries (Virginia)

“Spectral reciprocity and applications” (notes, video)

### [14-Feb-2022] Siegfred Baluyot (AIM)

“The shifted convolution of divisor functions” (notes, video)

## Fall 2021

“Averages of products and ratios of L-functions and characteristic polynomials” (video)

### [29-Nov-2021] Chung-Hang (Kevin) Kwan (Columbia)

“Some aspects of $$\Gamma$$-functions” (notes, video)

### [22-Nov-2021] Fan Ge (William & Mary)

“Critical points of the Riemann zeta-function” (notes, video)

### [15-Nov-2021] Fatma Çiçek (IIT Gandhinagar)

“Value distribution of the Riemann zeta-function on the half-line” (notes, video)

### [08-Nov-2021] Emilia Alvarez (Bristol)

“Random matrices and Painlevé equations” (video)

### [02-Nov-2021] Shehzad Hathi (UNSW Canberra)

“The LLL algorithm and its applications to Mertens conjecture” (notes, video)

### [25-Oct-2021] Max Xu (Stanford)

“Limiting distributions of sums of random multiplicative functions” (notes, video)

### [11-Oct-2021] Louis Gaudet (Rutgers)

“The Kloosterman Circle Method” (notes, video)

### [04-Oct-2021] Caroline Turnage-Butterbaugh (Carleton)

“Gaps between zeros of the Riemann zeta-function” (notes, video)

## Spring 2021

### [21-Jun-2021] Akshat Mudgal (Bristol/Purdue)

“An introduction to Vinogradov’s mean value theorem” (notes, video)

### [14-Jun-2021] Amita Malik (AIM/Max Planck)

“Equidistribution of $$\alpha p^{\theta}$$ with a Chebotarev condition and applications to extremal primes” (video)

### [07-Jun-2021] Winston Heap (Shandong)

“Mean values of long Euler products” (notes, video)

### [17-May-2021] Jared Duker Lichtman (Oxford)

“The Riemann hypothesis for curves” (notes, video)

### [19-Apr-2021] Vivian Kuperberg (Stanford)

“The second moment of quadratic twists of modular L-functions (after Soundararajan and Young)” (notes, video)

### [12-Apr-2021] Ofir Gorodetsky (Oxford)

“The variance of the sum of an arithmetic function over random short intervals” (notes, video)

### [08-Mar-2021] Max Xu (Stanford)

“Distribution and moments of random multiplicative functions” (notes, video)

### [22-Feb-2021] Chung-Hang (Kevin) Kwan (Columbia)

“Averages of Fourier coefficients” (notes, video)

### [15-Feb-2021] Quanli Shen (Lethbridge)

“The first moment of quadratic twists of modular L-functions” (notes, video)

### [08-Feb-2021] Emilia Alvarez (Bristol)

“An overview of random matrix theory for number theorists” (notes, video)

### [01-Feb-2021] Alessandro Fazzari (Genova)

“A survey on the central limit theorem for the Riemann zeta function” (notes, video)