July 18-22, 2022

Online number theory conference and school

Online number theory conference and school

Partitions is a branch of number theory initiated by Leonard Euler. After Euler the subject has been developed by many mathematicians such as Gauss, Jacobi, Schur, MacMahon, Hardy, Ramanujan, Andrews, Ono, etc. Mock theta functions, an important variants of theta functions, were discovered by the Indian mathematician Ramanujan in the early 20th century. The field of partitions is a proving ground where one tests the latest techniques from q-series and (mock) modularity. The topics of the conferences include but are not limited to the latest developments in q-series, partitions, modular forms, mock modular forms, and quantum modular forms.

Invited speakers

Bruce Berndt

University of Illinois, USA

George Andrews

Pennsylvania State University, USA

Nikolay Vavilov

Department of Mathematics and Computer Sciences, Saint Petersburg State University

Jeremy Lovejoy

Universite de Paris, France

Walter Bridges

University of Cologne, Germany

Ali Uncu

RICAM, Austria

Eric Mortenson

Department of Mathematics and Computer Sciences, Saint Petersburg State University

Atul Dixit

Indian Institute of Technology Gandhinagar

**CONFERENCE**

In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p(n), the number of partitions of n. In this talk, we shall provide a context for this result via MacMahon's Partition Analysis which leads directly to many other theorems of this nature, and which can be viewed as a continuation of our work on elongated partition diamonds. We find that generating functions which are infinite products built by the Dedekind eta function lead to interesting arithmetic theorems and conjectures for the related partition functions.

The little-known Russian mathematician, N. S. Koshliakov, introduced an analogue of the Riemann zeta function and an ingenious variant of the Abel–Plana summation formula that arise from a boundary value problem in heat conduction. Connections can be made with analogues of the Abel–Plana summation formula found in Ramanujan’s notebooks. Koshliakov also found special cases of arithmetical series identities involving infinite series of K-Bessel functions. New more general theorems along these lines are discussed. Most of this lecture is related to joint research with Atul Dixit, Rajat Gupta, and Alexandru

Zaharescu.

Cylindric partitions are a sort of shifted repeating plane partition, and they always have infinite product generating functions. Corteel and Welsh recently developed machinery that one can use to look for a corresponding sum-side and hence a product-sum identity. The product-sides for cylindric partitions are rather limited, but here we show that Corteel and Welsh's work may be extended to weighted cylindric partitions, among other objects, which allow for essentially any product-side. We give new proofs of the little G\"{o}llnitz and G\"{o}llnitz--Gordon identities, some ``Schmidt-type'' identities and a few novel identities. This is joint work with Ali Uncu.

Lambert series lie at the heart of q-series, modular forms and the theory of the Riemann zeta function. Among the pioneers in the subject were Ramanujan and Wigert. We will discuss Ramanujan’s formula for odd zeta values involving the Lambert series associated to n^k where k is is an odd integer, not equal to −1. Generalizations and analogues of this formula obtained by the speaker with his co-authors over the past few years will be presented. The second half of the talk is concerned with a transformation of the Lambert series of logarithm recently found by Soumyarup Banerjee, Shivajee Gupta and the author. The complete asymptotic expansion of this series, as q → 1, will be derived. Its application, in turn, in the zeta-function theory will then be given.

Zagier introduced the term ``strange identity" to describe an asymptotic relation between a certain q-hypergeometric series and a partial theta function at roots of unity. We explain how Zagier's strange identity may be viewed from the perspective of Bailey pairs. Using the iterative machinery of Bailey pairs then leads to many families of multisum strange identities, including Hikami's generalization of Zagier's identity.

We present the basic theory of harmonic and locally harmonic Maaß forms. In addition, we discuss some classical examples of each class of forms. We highlight that locally harmonic Maaß forms can be utilized to detect the vanishing of certain twisted central L values. To this end, we follow the papers by Bringmann, Kane, Kohnen, and by Ehlen, Guerzhoy, Kane, Rolen.

The cylindric partitions defined by Gessel and Krattenthaler have been of recent interest after a paper by Corteel and Welsh. In this talk, we will present the necessary background to the topic and my (joint with Corteel and Dousse) recent contribution to this subject. We will also mention some newer developments and talk about prospects.

The talk is based on a series of my survey papers on the role of computers in number theory. I will briefly sketch the current status of Waring and easier Waring problems, Goldbach problem, Mersenne---Catalan problem, factorisation of Fermat numbers, and the like, and state them as computer algebra challenges. Andre Weil started his classical text on the history of number theory with “The main thesis will be the continuity of number theory for the last three hundred years and the fact that what we are doing now is in direct continuation of what has been done by the greatest number-theorists since Fermat started it all in the seventeenth century”. I cannot agree more. I wish to demonstrate how computer algebra systems allow us to reconnect to the history of mathematics as a whole, not just the conceptual mathematics of the XIX and XX centuries, but also to the more computationally oriented XVII and XVIII century mathematics. As everyone knows, Ramanujan was in direct contact with the goddess Namakkal. However, for a long time most mathematicians lacked similar facility. Nowadays, Mathematica and Maple easily breed formulas in the same style as Euler and Ramanujan did. This lifts some of the mystery, but on the other hand, restores the balance between ideas and calculations that was seriously upset during a large part of the XX century.

**SCHOOL**

School Lecture 1: Partition bijections I: Euler's odds-distinct theorem and beyond

School Lecture 2: Partition bijections II: the involution principle and O'Hara's algorithm

School Lecture 3: Partition inequalities: Ehrenpreis's problem, motivated proofs and the anti-telescoping method

School Lecture 1: Introduction to Partitions, Generating Functions and Partition Identities.

School Lecture 2: Double-sum, multi-sum generating functions for partitions from a combinatorial point of view.

School Lecture 3: Some refinements and reflections of multi-sum generating functions and implications.

School Lecture 1: Waring problem and easier Waring problem

School Lecture 2: Mesrenne and Fermat numbers, and divisor sums

School Lecture 3: Goldbach hypothesis and distribution of primes

Schedule

CET (Central European/Berlin Time)

Conference Day1

Monday, 18 July

Monday, 18 July

16.00-16.45 (CET) Bruce Berndt

17.00-17.45(CET) George Andrews

18.00-18.45(CET) Nikolai Vavilov

17.00-17.45(CET) George Andrews

18.00-18.45(CET) Nikolai Vavilov

Conference Day2

Tuesday,19 July

Tuesday,19 July

15.00-15.45(CET) Jeremy Lovejoy

16.00-16.45 (CET) Atul Dixit

17.00-17.45(CET) Ali Uncu

18.00-18.45(CET) Walter Bridges

16.00-16.45 (CET) Atul Dixit

17.00-17.45(CET) Ali Uncu

18.00-18.45(CET) Walter Bridges

School Day 1

Wednesday, 20 July

Wednesday, 20 July

15.00-16.15(CET) Nikolai Vavilov

16.30-17.45 (CET) Ali Uncu

18.00 - 19.15 (CET) Walter Bridges

16.30-17.45 (CET) Ali Uncu

18.00 - 19.15 (CET) Walter Bridges

School Day 2

Thursday, 21July

Thursday, 21July

15.00-16.15(CET) Nikolai Vavilov

16.30-17.45 (CET) Ali Uncu

18.00 - 19.15 (CET) Walter Bridges

16.30-17.45 (CET) Ali Uncu

18.00 - 19.15 (CET) Walter Bridges

School Day 3

Friday, 22 July

Friday, 22 July

15.00-16.15(CET) Nikolai Vavilov

16.30-17.45 (CET) Ali Uncu

18.00 - 19.15 (CET) Walter Bridges

19.30 - 20.15 (CET) Andreas Mono

16.30-17.45 (CET) Ali Uncu

18.00 - 19.15 (CET) Walter Bridges

19.30 - 20.15 (CET) Andreas Mono

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