Illinois Number Theory Seminar : Spring 2018

Nicolas Robles (University of Illinois)

Combinatorial aspects of Levinson's method

The celebrated theorem of Levinson (1974) states that more than 1/3 of the non-trivial zeros of the Riemann zeta-function are on the critical line. This result has been improved during the last 40 years by employing linear and first order terms of a mollifier as well as by using Kloostermania techniques for the error terms.

In this work, we delineate how to improve all degrees of the most natural and powerful Dirichlet series (producing an arbitrarily perfect mollification) and we also present the best error terms available with our current technology of exponential sums by elucidating a conjecture of S. Feng. A new and modest % record is thereby achieved.

Joint work with Kyle Pratt, Alexandru Zaharescu and Dirk Zeindler.

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Harold Diamond (University of Illinois)

The convolution square root of  1  and application to the prime number theorem.

We explain what this arithmetic function is and show how the PNT can be deduced from knowledge of its summatory function.

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Byungchan Kim (Seoul Technical University)

The theta-operator on eta-quotients

Eta-quotients are a important explicit class of modular forms, and the theta-operator is a natural operator on modular forms. We investigate which eta-quotients are preserved by the theta-operator. This is motivated by a particular partition congruence. This talk is based on joint work with P.-C. Toh and with D. Choi and S. Lim.

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Patrick Allen (Illinois)

Potential automorphy and applications

The philosophy of Langlands reciprocity predicts that many L-functions studied by number theorists should be equal to L-functions coming from automorphic forms. This leads to Langlands's functoriality conjecture, which very roughly states L-functions naturally created from a given automorphic L-function should also be automorphic. I'll describe this in the case of symmetric power L-functions and how the Langlands program in this special case has applications to the Sato-Tate conjecture and the Ramanujan conjecture. The former concerns the distribution of points on elliptic curves modulo various primes, while the latter concerns the size of the Fourier coefficients of modular forms. I'll then discuss joint work in progress with Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne establishing a weak form of Langlands reciprocity and functoriality for symmetric powers of certain rank 2 L-functions over CM

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Corey Stone (Illinois)

Euler Systems and Special Values of L-functions

In the 1990s, Kolyvagin and Rubin introduced the Euler system of Gauss sums to derive upper bounds on the sizes of the p-primary parts of the ideal class groups of certain cyclotomic fields. Since then, this and other Euler systems have been studied in order to analyze other number-theoretic structures.

Recent work has shown that Kolyvagin’s Euler system appears naturally in the context of various conjectures by Gross, Rubin, and Stark involving special values of L-functions.  We will discuss these Euler systems from this new point of view as well as a related result about the module structure of various ideal class groups over Iwasawa algebras.

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Paulina Koutsaki (Illinois)

 Monotonicity properties of L-functions

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Xianchang Meng (McGill)

Some generalizations of prime number races problems

Chebyshev observed that there seem to be more primes congruent to 3 mod 4 than those congruent to 1 mod 4, which is known as Chebyshev’s bias. In this talk, we introduce two generalizations of this phenomenon. 1) Greg Martin conjectured that the difference of the summatory function of the number of prime factors over integers <x from different arithmetic progressions will attain a constant sign for sufficiently large x. Under some reasonable conjectures, we give strong evidence to support this conjecture. 2) We introduce the function field version of Chebyshev’s bias. We consider the distribution of products of irreducible polynomials over finite fields. When we compare the number of such polynomials among different arithmetic progressions, new phenomenon will appear due to the existence of real zeros for some associated L-functions.

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Bogdan Petrenko (Eastern Illinois U.)

Some conjectural properties of coefficients of cyclotomic polynomials

The goal of this talk is to interest the audience in some  puzzling experimental observations about the asymptotic behavior of coefficients of cyclotomic polynomials.It is well known that any integer is a coefficient of some cyclotomic polynomial.We find it intriguing that when various families of coefficients of cyclotomic polynomials are plotted on the computer screen, the resulting pictures appear"asymptotically almost symmetric".At present, we do not have any theoretical explanationof this perceived behavior of the coefficients. This talk is based on my joint workin progress with  Brett Haines  (WolframResearch),Marcin Mazur (Binghamton University),and William Tyler Reynolds (University of Iowa).

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Jeremy Lovejoy (Paris 7)

Colored Jones polynomials and modular forms

In this talk I will discuss joint work with Kazuhiro Hikami, in which we use Bailey pairs and the Rosso-Jones formula to compute the cyclotomic expansion of the colored Jones polynomial of a certain family of torus knots.   As an application we find quantum modular forms dual to the generalized Kontsevich-Zagier series. As another application we obtain formulas for the unified WRT invariants of certain 3-manifolds, some of which are mock theta functions.    I will also touch on joint work with Robert Osburn, in which we compute a formula for the colored Jones polynomial of double twist knots.

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Fernando Shao (Univ. Kentucky)

Around Vinogradov's three primes theorem

Vinogradov showed in 1937 that every large enough odd integer can be represented as a sum of three primes. One may ask what if these primes are restricted to some (potentially sparse) subset of the primes. In general, if the set is badly distributed in congruence classes or Bohr sets, the result does not necessarily hold. In this talk I will describe two "transference type'' results aimed to show the obstructions described above are the only obstructions. As applications, we get that Vinogradov's three primes theorem holds for Chen primes and for primes in short intervals. This is based on joint works with K. Matomaki and J. Maynard.

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Kevin Ford (University of Illinois)

Large gaps in sieved sets

For each prime , remove from the set of integers a set of residue classed modulo p, and let S be the set of remaining integers.   As long as has average 1, we are able to improve on the trivial bound of , and show that for some positive constant c, there are gaps in the set S of size as long as x is large enough.   As a corollary, we show that any irreducible polynomial f, when evaluated at the integers up to X, has a string of consecutive composite values, for some positive c (depending only on the degree of f).   Another corollary is that for any polynomial f, there is a number G so that for any , there are infinitely many values of n for which none of the values are coprime to all the others.  For , this was proved by Erdos in 1935, and currently it is known only for linear, quadratic and cubic polynomials. This is joint work with Sergei Konyagin, James Maynard, Carl Pomerance and Terence Tao.

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George Shakan (Illinois)

Higher order energy decompositions and the sum-product phenomenon.

In 1983, Erdos and Szemeredi conjectured that either |A+A| or |AA| is at least , up to a power loss. We make progress towards this conjecture by using various energy decomposition results, in a similar spirit to the recent Balog-Wooley decomposition. Our main tool is the Szemeredi-Trotter theorem from incidence geometry. For more information, see my blog which contains a video introduction the subject: gshakan.wordpress.com

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Junxian Li and A. J. Hildebrand (Illinois)

The Unreasonable Effectiveness of Benford's Law in Mathematics

We describe work with Zhaodong Cai, Matthew Faust, and Yuan Zhang that originated with some unexpected experimental discoveries made in an Illinois Geometry Lab undergraduate research project back in Fall 2015. Data compiled for this project suggested that Benford's Law (an empirical "law" that predicts the frequencies of leading digits in a numerical data set) is uncannily accurate when applied to many familiar mathematical sequences. For example, among the first billion Fibonacci numbers exactly 301029995 begin with digit 1, while the Benford prediction for this count is 301029995.66. The same holds for the first billion powers of 2, the first billion powers of 3, and the first billion powers of 5. Are these observations mere coincidences or part of some deeper phenomenon? In this talk, which is aimed at a broad audience, we describe our attempts at unraveling this mystery, a multi-year research adventure that turned out to be full of surprises, unexpected twists, and 180 degree turns, and that required unearthing nearly forgotten classical results as well as drawing on some of the deepest recent work in the area.

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Kyle Pratt (Illinois)

Average non-vanishing of Dirichlet L-functions at the central point

 One expects that an L-function vanishes at the central point either for either deep arithmetic reasons, or for trivial reasons. The central values of Dirichlet L-functions have no arithmetic content, and are also not forced to vanish by the functional equation. One is then led to believe that these central values never vanish, which is a conjecture going back in one form or other to Chowla. The Generalized Riemann Hypothesis implies that almost half of these central values are nonzero. In this talk I will discuss my recent work on central values of Dirichlet L-functions. The main theorem, an unconditional result, is beyond the reach of the Generalized Riemann Hypothesis.