### 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.

• December 7,2017: Daniel Elbaz (TAU) How many prime divisors do integers have?

Abstract: A hundred years ago, Hardy and Ramanujan proved that the typical integer n has about log log n prime divisors. In 1940, Erdos and Kac established a remarkable central limit theorem for the number of prime divisors. We will explain what that means and present Billingsley's proof (1969) of the Erdos-Kac theorem.

• December 14, 2017: No seminar today. Happy Hanukka!
• December 17, 2017 (Sunday): The VIth Israeli Algebra and Number Theory Day (in Jerusalem)
• December 21, 2017: Zeev Rudnick (TAU). The least common multiple of polynomial sequences.

Abstract: Chebyshev investigated the least common multiple of the first N integers, showing that its behaviour is closely related to the Prime Number Theorem. In the lecture, I will discuss a conjecture of Cilleruelo about the least common multiple of polynomial sequences.

• December 28, 2017: Igor Wigman (KCL). The directional distribution of nodal lines.
• January 4, 2018: Michael Magee (Durham). Uniform spectral gap in number theory (CANCELLED)

Abstract: I'll begin by discussing Selberg's eigenvalue conjecture. This is an analog of the Riemann hypothesis for a special family of Riemann surfaces that feature heavily in number theory, for example in Wiles' proof of the Taniyama-Shimura conjecture. I'll explain how in the last 10-15 years, number theorists have had to turn to Anosov dynamics to obtain the approximations to Selberg's conjecture that became relevant to emerging 'thin groups' questions about Apollonian circle packings and continued fractions. I will explain the spectral gap results I worked on in this area. Then if I have time, I'll explain how I am now looking for analogs of the Selberg conjecture in the setting of Teichmuller dynamics with yet more interesting number theory questions in mind.

• January 11, 2018: Efrat Bank (Michigan) Correlation between primes in short intervals on curves over finite fields.

Abstract: In this talk, I present an analogue of the Hardy-Littlewood conjecture on the asymptotic distribution of prime constellations in the setting of short intervals in function fields of smooth projective curves over finite fields. I will discuss the definition of a "short interval" on a curve as an additive translation of the space of global sections of a sufficiently positive divisor E by a suitable rational function f, and show how this definition generalizes the definition of a short interval in the polynomial setting. I will give a sketch of the proof which includes a computation of a certain Galois group, and a counting argument, namely, Chebotarev density type theorem. This is a joint work with Tyler Foster.

• January 18, 2018: Alex Gamburd (CUNY) Arithmetic and Dynamics on Markoff-Hurwitz Varieties

Abstract: Markoff triples are integer solutions to Markoff equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these (and, in particular, detailing the relation to product replacement algorithm) we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected (numerical evidence indicates that they are, in fact, expanders). Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite. Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.

• January 25-March 1 no seminar (break between semesters).
• March 8, 2018 Patrick Meisner (TAU). Erdos' multiplication table problem
• March 15, 2018: Jens Marklof (Bristol). Higher dimensional Steinhaus problems

Abstract: The three gap theorem, also known as Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of a,2a,...., Na, take at most three distinct values. Motivated by a question of Erdos, Geelen and Simpson, we explore a higher-dimensional variant, which asks for the number of gaps between the fractional parts of a linear form. Using the ergodic properties of the diagonal action on the space of lattices, we prove that for almost all parameter values the number of distinct gaps in the higher dimensional problem is unbounded. This in particular improves earlier work by Boshernitzan, Dyson and Bleher et al. Joint work with Alan Haynes (Houston).

• March 22, 2018: Shuai Zhai (Cambridge). The congruent number problem and the Birch and Swinnerton-Dyer conjecture

Abstract: In this lecture, I will present the congruent number problem, which is the oldest major unsolved problem in number theory and can be traced back at least to the 10th century in Arab manuscripts, and its relation to the conjecture of Birch and Swinnerton-Dyer.

• March 29, April 5, 2018 - no seminar (Passover break).
• April 12, 2018 Lior Rosenzweig (Afeka college), Prime and Mobius correlations for very short intervals in polynomials over finite fields.

Abstract: We investigate function field analogs of the distribution of primes, and prime k-tuples, in "very short intervals", as well as cancellation in sums of function field analogs of the Mobius function and its correlations. When the characteristic of the field tends to infinity, we show that for "generic" polynomials the main term agrees with the expected results in the classical setting. We also give examples of polynomials for which there is no cancellation at all, and intervals where the heuristic "primes are independent" fails badly. This is joint work with Par Kurlberg.

• April 19, 2018 no seminar (Independence day).
• April 26, 2018. Ezra Waxman (TAU): Variance of Gaussian Primes across Sectors

Abstract: A Gaussian prime is a prime element in the ring of Gaussian integers Z[i]. As the Gaussian integers lie on the plane, interesting questions about their geometric properties can be asked which have no classical analogue among the ordinary primes. Specifically, to each Gaussian prime a + bi, we may associate an angle whose tangent is the ratio b/a. Hecke proved that these Gaussian primes are equidistributed across sectors of the complex plane, by making use of (infinite) Hecke characters and their associated L-functions. In this talk I will present a conjecture, motivated by a random matrix model, for the asymptotic variance of Gaussian primes across sectors. I will also discuss ongoing work towards a more refined conjecture, which picks up lower-order-terms. Finally, I will introduce a function field model for this problem, which will yield an analogue to Hecke's equidistribution theorem. By applying a result of N. Katz concerning the equidistribution of "super even" characters (the function field analogues to Hecke characters), I will provide a result for the variance of function field Gaussian primes across sectors; a computation whose analogue in number fields is unknown beyond a trivial regime (Joint work with Zeev Rudnick)

• May 3, 2018 Bingrong Huang (TAU): Prime lattice points in ovals.

Abstract: We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta functions are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.

• May 10, 2018 Barak Weiss (TAU), Recent developments around Minkowski's conjecture

Abstract: Minkowski's conjecture is a longstanding open problem in the geometry of numbers. It states that for every lattice L of covolume 1 in R^n, and for every x in R^n, there is y in L such that the absolute value of the product of the differences of coefficients of x and y is bounded above by 2^{-n}. Over the years and through work of many mathematicians, this statement has been verified up to dimension 9. It remains open in higher dimensions. Recent work about the conjecture has explored connections with dynamics on homogeneous spaces, dimension theory, and asymptotic geometry of Banach spaces. I will survey this recent work and explain the current state of our knowledge.

• May 17, 2018 Sanghoon Kwon, Kwandong University, Korea. A combinatorial approach to the Littlewood conjecture in positive characteristic.

Abstract: The Littlewood conjecture is an open problem in simultaneous Diophantine approximation of two real numbers. Similar problem in a field K of formal series over finite fields is also still open. This positive characteristic version of problem is equivalent to whether there is a certain bounded orbit of diagonal semigroup action on Bruhat-Tits building of PGL(3,K). We describe geometric properties of buildings associated to PGL(3,K), explore the combinatorics of the diagonal action on it and discuss how it helps to investigate the conjecture.

• May 24, 2018 no meeting ( Annual meeting of the Israel Math Union, Technion, Haifa)
• May 31, 2018 (possible conflict with TAU student day).
• June 7, 2018 No meeting ( Perspectives on the Riemann Hypothesis, Bristol)

### Homework Assignments

1. Assignment 1 , due Nov 9.
2. Assignment 2 , due Nov 16, 2017.
3. Assignment 3 , due January 4, 2018

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