Frank Patane

An identity connecting theta series associated with binary quadratic forms of discriminant Delta and Delta p^2

When/Where:

October 7, 2014, at 1:55pm, in LIT 368

Abstract:

I will state and prove a new identity which connects theta series associated with binary quadratic forms of idoneal discriminants \( \Delta\) and \(\Delta p^2\), for \(p\) a prime. I will then illustrate how to use this identity to derive Lambert series identities and hence product representation formulas for certain forms. Last but not least, I discuss generalizations to non-idoneal discriminants and the resulting theta series identities one may derive.

Alexander Berkovich

Some implications of the 1907 Hurwitz formula II

When/Where:

September 16, 2014, at 1:55pm, in LIT 368

Abstract:

Let \(s(n)\) denote \(\#\) of representations of integer \(n\) by \(x^2+y^2+z^2\). Let \(n\) be odd positive integer. In my first talk I showed that \(s(n^2)\geq 6n\) with equality if and only if all prime divisors of \(n\) are congruent to \(1 \pmod{4}\). I employ the above inequality and a special case of the Jacobi triple product identity to prove certain conjectures of Kaplansky. In particular, I will show that \(4x^2 + 9y^2 + 32z^2 +4xy \) represents, exclusively, all positive integers not of the form

  • \(4^a(4m+2), a=0,1 \)
  • \(4^a(8m+3), a=0,1 \)
  • \(4^a(8m+5), a=0,1 \)
  • \(4^a(8m+7), a\geq 0,\)
  • or \(M^2 \)

where \(a,m,M\) are non-negative integers and \(M\) is generated by \(1\) and primes congruent to \(1 \pmod{4} \).

Alexander Berkovich

Some implications of the 1907 Hurwitz formula

When/Where:

September 9, 2014, at 1:55pm, in LIT 368

Abstract:

I start with a brief review of Siegel’s formula for the number of representations of a positive integer by a genera of a quadratic form. I show how this formula implies the 1907 observation by Hurwitz, who proposed that \(\left|\{ (x,y,z) \in \mathbb{Z}^3 \mid n^2 = x^2+y^2+z^2 \}\right|\) may be expressed as a simple finite function of the divisors of \(n \in \mathbb{N}\). I use this formula together with the Jacobi triple product identity to prove (among other things) that \(9x^2+ 16y^2 +36z^2 + 16yz+ 4xz + 8xy\) represents exclusively all positive integers not of the form

  • \(4^a(8m+7)\);
  • \(4^a(8m+3)\), \(a=0,1,2\);
  • \(4^a(4m+2)\), \(a=0,1,2\);
  • \(4^a(8m+5)\), \(a=0,1\);
  • \(M^2\);
  • or \(4M^2\);

where \(a,m,M\) are non-negative integers and \(M\) is generated by 1 and primes congruent to \(1\pmod{4}\).

George Andrews

Identities missed by Schur and MacMahon

When/Where:

April 22, 2014, at 1:55pm, in LIT 368

Abstract:

The first portion of the talk will describe a new formulation for the generating function for the partitions in Schur’s 1926 partition theorem (joint work with Kathrin Bringmann and Karl Mahlburg). If there is time, I will describe how MacMahon “just missed” proving the Rogers-Ramanujan identities.

Duc Huynh

Chebotarev Density Theorem – its history and applications

When/Where:

April 15, 2014, at 1:55pm, in LIT 368

Abstract:

The Chebotarev Density Theorem states that the frequency of the occurrence of a given splitting pattern of prime ideals, for all primes \(p\) less than a large integer \(N\), tends to a certain limit as \(N\) goes to infinity. It generalizes Dirichlet’s theorem on arithmetic progressions and the Frobenius Density Theorem. The work of Chebotarev led the way to the Artin Reciprocity Theorem. The theorem yields many important applications including the reduction of the problem of classifying Galois extensions of a number field to that of describing the splitting of primes in extensions.

We will discover the history of the theorem from Gauss Quadratic Reciprocity to Artin Reciprocity Law. We will learn of the struggles and brilliance of Nikolai Chebotarev. Finally, we will catch a glimpse of Chebotarev’s counterintuitive proof to his density theorem.

“I belong to the old generation of Soviet scientists, who were shaped by the circumstance of a civil war. I devised my best result while carrying water from the lower part of town to the higher part, or buckets of cabbages to the market, which my mother sold to feed the entire family.” – Nikolai Chebotarev

Keith Grizzell

Generalization and refinement of the Berkovich-Garvan partition inequalities

When/Where:

April 1, 2014, at 1:55pm, in LIT 368

Abstract:

In section 5 of their paper “Dissecting the Stanley Partition Function”, Profs. Berkovich and Garvan showed how to prove an infinite collection of new partition inequalities by constructing a (non-trivial) injection. I will review these inequalities and the injection they used. Then, I will look at generalizing and refining both the inequalities and the injection, in a manner akin to what Prof. Berkovich and I did in our recent paper “A partition inequality involving two \(q\)-Pochhammer symbols”. Finally, I will discuss how this method of generalization should also work in some other settings.

Ae Ja Yee

The truncated Jacobi triple product theorem

When/Where:

March 17, 2014, at 1:55pm, in LIT 339 (the Atrium)

Abstract:

Recently, G. E. Andrews and M. Merca considered the truncated version of Euler’s pentagonal number theorem and obtained a non-negativity result. They asked the same question on the truncated Jacobi triple product identity. In this talk we will discuss this question.

Christopher Jennings-Shaffer

Higher Order Overpartition Spt Functions

When/Where:

April 8, 2014, at 1:55pm, in LIT 368

Abstract:

In 2003 Atkin and Garvan introduced the \(k\)-th rank and crank moments \(M_k(n) = \displaystyle\sum_{m}m^k M(m,n)\) and \(N_k(n) = \displaystyle\sum_{m}m^k N(m,n)\), where \(N(m,n)\) is the number of partitions of \(n\) with rank \(m\) and \(M(m,n)\) is the number of partitions of \(n\) with crank \(m\). A symmetrized version of the rank moments was studied by Andrews in 2006. In 2011 Garvan used symmetrized rank and crank moments to define a higher order \(\mathrm{spt}\) function, noting Andrews’ \(\mathrm{spt}\) function is half the difference of the second crank and rank moment. This also established an inequality between crank and rank moments.

We show this idea extends to the \(\mathrm{spt}\) functions for overpartitions, overpartitions with smallest part even, and partitions with smallest part even and without repeated odd parts. In this talk we give the necessary definitions, in particular what are the symmetrized moments, and review the steps for the ordinary \(\mathrm{spt}\) function. We then continue on with the three additional \(\mathrm{spt}\) functions. The crucial identities follow from a theorem in Garvan’s work that is an application of Bailey pairs and Bailey’s Lemma.

Peter Paule

Ramanujan’s Congruences Modulo Powers of 5, 7, and 11 Revisited

When/Where:

March 18, 2014, at 1:55pm, in LIT 368

Abstract:

The number of partitions of \(4\) is \(p(4)=5\), namely: \(4\), \(3+1\), \(2+2\), \(2+1+1\), and \(1+1+1+1\). Ramanujan observed that \(p(5n+4)\) is divisible by \(5\) for all non-negative integers \(n\). More generally, Ramanujan discovered similar congruences modulo \(7\) and \(11\), and also for all powers of these primes. The cases \(5\) and \(7\) were proved by G.N. Watson (1938); in 1984, Frank Garvan was able to simply this proof significantly. In 1967 the case \(11\) was proved by A.O.L. Atkin; in 1983, B. Gordon presented another approach being closer to Watson’s. The talk originates from joint work with Silviu Radu; it describes a new algorithmic setting in the theory of modular functions that gives rise to a new unified frame to prove Ramanujan’s celebrated families of partition congruences.

George Andrews

Bressoud’s “Easy Proof of the Rogers-Ramanujan Identities” and some unexpected consequences

When/Where:

March 11, 2014, at 1:55pm, in LIT 368

Abstract:

A multiple series generalization of the Rogers-Ramanujan identities (analytic form) was given in 1974. We begin this talk with a brief discussion of that result and its proof. Then in 1983, David Bressoud published: “An easy proof of the Rogers-Ramanujan identities,” and the first half of the paper is exactly that. The paper concludes with a multiple series generalization that has unexpected consequences. We shall examine the Bressoud paper in light of the Bailey chain machinery and examine the unexpected consequences.