Prime number conjectures from the Shapiro class structure

When/Where:

January 22, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

The height \(H(n)\) of \(n\), introduced by Pillai in 1929, is the smallest positive integer \(i\) such that the \(i\)-th iterate of Euler’s totient function at \(n\) is 1. H. N.Shapiro (1943) studied the structure of the set of all numbers at a particular height. We provide a formula for the height function thereby extending a result of Shapiro. We list steps to generate numbers of any height which turns out to be a useful way to think about this construct. We present some theoretical and computational evidence to show that \(H\) and its relatives are closely related to the important functions of number theory, namely \(\pi(n)\) and the \(n\)-th prime \(p_n\). This is joint work with Hartosh Singh Bal.

Sudler’s Products. Some New Observations

When/Where:

April 17, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

I will discuss my latest joint work with Ali Uncu (RISC JKU)

The Combinatorics of the leading root of Ramanujan’s function

When/Where:

March 20, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

I consider the leading root \(x_0(q)\) of Ramanujan’s function (or \(q\)-Airy function) \(\sum\limits_{n=0}^\infty\frac{(-x)^nq^{n^2}}{(1-q)(1-q^2)\ldots(1-q^n)}\). I prove that its formal power series expansion

\(qx_0(-q)=1+q+q^2+2q^3+4q^4+8q^5+\ldots\)

has positive integer-valued coefficients, by giving an explicit combinatorial interpretation of these numbers in terms of trees whose vertices are decorated with polyominos.

Similar results are also obtained for the leading roots of the partial Theta function and the Painleve Airy function.

Higher-order Multi-critical Points in Two-dimensional Lattice Polygon Models

When/Where:

March 13, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, “jumps” orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half length, area and number of jumps. This represents the first example of a hierarchy of exact solvable two-dimensional lattice vesicle models showing higher-order multi-critical points with scaling functions expressible via generalized Airy functions, as conjectured bu John Cardy.

Basic hypergeometric expressions for q-Tangent and q-Secant Numbers

When/Where:

February 27, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

We derive new expressions for the generating functions of q-tangent and q-secant numbers from enumerating path diagrams given by Dyck paths together with partial fillings below the paths. In doing so, we provide expressions for path diagrams with restricted height involving basic hypergeometric functions, obtained by solving recurrences arising from the continued fraction representation of the generating functions.

Revisiting the Riemann Zeta Function at the Positive Even Integers

When/Where:

February 20, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

Euler showed that the values of the Riemann zeta function at positive even integer arguments \(2k\) are rational multiples of \(\pi^{2k}\), these rationals being given in terms of Bernoulli numbers. Over the years, several proofs of this celebrated result of Euler have been given. We will discuss a new proof by simply starting with the determination of the Fourier coefficients of \(f(x)=x^k\), and using the Parseval identity. This leads to a pair of intertwining recurrences, which when investigated closely leads to a very different proof of Euler’s formula and a surprising new identity for Bernoulli numbers.

This is joint work with Colin Defant. Time permitting,

I will also discuss connections between Bernoulli polynomials and the analytic continuation of the Riemann zeta function.

Higher Order Mock Theta Conjectures

When/Where:

February 13, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

The Mock Theta Conjectures were identities stated by Ramanujan for his so called fifth order mock theta functions. Andrews and the speaker showed how two of these fifth order functions are related to rank differences mod 5. Hickerson was first to prove these identities and was also able to relate the three Ramanujan seventh order mock theta functions to rank differences mod 7. Based on work of Zwegers, Zagier observed that the two fifth order functions and the three seventh order functions are holomorphic parts of real analytic vector modular forms on \(SL_2(Z)\). Zagier gave an indication how these functions could be generalized. We give details of these generalizations and show how Zagier’s 11th order functions are related to rank differences mod 11.

Problems from David Newman

When/Where:

February 6, 2018, 3:00 — 3:50 pm at LIT 368.

Abstract:

David Newman has number theory as his hobby. He and I have corresponded for more than 20 years. Recently we have written two joint papers. The first concerns special, non-standard infinite product representations of classical theta functions. The second concerns partition problems related to Fraenkel’s MEX function in partitions. MEX = MinimalEXcludant.

\((1-q)(1-q^2)\dots(1-q^m)\) Revisited

When/Where:

October 31, 2017, 3:00 — 3:50 pm at LIT 368.

Abstract:

 Let    
\(\displaystyle (1-q)(1-q^2) … (1-q^m)  = \sum_{n>= 0}  c_m (n) q^n.\)

In this talk I discuss how to use \( q\)- binomial theorem  together with   the Euler pentagonal number theorem to show that  \(\max( |c_m|) = 1\)  iff \( m = 0,\ 1,\ 2,\ 3,\ 5\).   There are many other similar results.  For example, it can be  proven  that no positive integer m exists such that \(\max( |c_m|) = 9\).

This talk is based on my recent joint work with  Ali Uncu  (RISC, Linz).

Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

When/Where:

October 24, 2017, 3:00 — 3:50 pm at LIT 368.

Abstract:

In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has been proved for degrees \(d\leq3\). We obtain an arbitrary precision asymptotic formula for the derivatives \(\Xi^{(2n)}(0)\), which allows us to prove the hyperbolicity of \(100\%\) of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This general condition also confirms a conjecture of Chen, Jia, and Wang.