### Organiser: Zeev Rudnick

The seminar will discuss topics in current research in number theory. Presentations will be by students, locals and visitors.

### Schedule

• October 26, 2017. Zeev Rudnick (TAU), The Erdos-Mirsky problem

Abstract: The frequency of occurrence of locally repeated" values of arithmetic functions is a common theme in analytic number theory. For instance, the Erdos-Mirsky problem asked to show that there are infinitely many integers n for which d(n)=d(n+1), where d(n) is the divisor function; and the number of such occurrences was the subject of a series of papers of Erdos with Pomerance and Sarkozy. Replacing d(n) by the Euler totient function leads to somewhat different problems. After describing the state of the art on these questions, I will introduce the corresponding problems in the setting of polynomials over a finite field, and the connection to the theory of random permutations.

• November 2, 2017. Bingrong Huang (TAU), Zero density theorems

Abstract: Studying zeros of L-functions is important in analytic number theory. For example, the Prime Number Theorem follows from a zero free region of the Riemann zeta function. The Riemann Hypothesis tells us that all non-trivial zeros are on the critical line and has a lot of consequences, but for now it is open. Substitutes for the Riemann Hypothesis are "zero density theorems", which bound the number of zeros off the critical line. In this talk, I will give some zero density theorems of the Riemann zeta function (Montgomery and Huxley) and the application to primes in short intervals.

• November 9, 2017 Alexei Entin (TAU). Decomposition statistics over function fields, a geometric view.

Abstract: we show how many recent results concerning the decomposition statistics of polynomials and divisors over F_q in the q->infinity limit can be approached through a unified geometric framework. This allows to easily reproduce, strengthen and generalize these results.

• November 16, 2017 Par Kurlberg (KTH, Stockholm). Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers.

Abstract: We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime k-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rudnick and Ueberschaer, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. We also give numerical evidence for the conjecture and its implications. Time permitting, we will sketch the proof of a key technical result, namely that a certain average over Hardy-Littlewood constants equals one.

• November 23, 2017: Bingrong Huang (TAU). Zero density theorems: The proofs.
• November 30, 2017: Mark Shusterman (TAU). Prime numbers (and polynomials) in shorter intervals

Abstract: The Riemann Hypothesis implies that every short interval centered at X of length slightly greater than X^{1/2} contains a prime. Using zero density theorems, one can get unconditional results with somewhat longer intervals. On the other hand, it is widely believed that the exponent 1/2 can be replaced by an arbitrarily small constant. Amazingly, finding a number with at most two prime factors in a short interval turns out to be much easier than finding a prime. The difficulty lies in guaranteeing a number with an odd (or with an even) count of prime factors. In this talk we consider the analog for prime polynomials over a finite field (instead of prime numbers), and show that in shorter intervals, one can still break the parity barrier mentioned above.

• January 11, 2018: Efrat Bank (Michigan)
• January 25, 2018: Alex Gamburd (CUNY) Arithmetic and Dynamics on Markoff-Hurwitz Varieties
• March 15, 2018: Jens Marklof (Bristol)
• March 22, 2018: Shuai Zhai (Cambridge)

### Homework Assignments

1. Assignment 1 , due Nov 9.
2. Assignment 2 , due Nov 16

Contact me at: rudnick@post.tau.ac.il, Office : Schreiber 316, tel: 640-7806