Two new expansions for partial sums of Gauss’ triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function p(n) and for the partition function pod(n) counting the partitions of n with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.

A new expansion is given for partial sums of Euler’s pentagonal number series. As a corollary we derive an infinite family of inequalities for the partition function, p(n).

I construct a complete 1-loop partition function of a bosonic closed string on orbifolds. Furthermore, I derive sufficient conditions for the modular invariance of the partition function.

We show that a partition function on the not-flat D1-brane can be written in the same form as that on the flat one in α-order. In this case the information of the curvature of the brane configuration is included in tachyon beta function.

For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2×2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived.

Abstract. We find a local (d + 1)× (d + 1) Riemann-Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of the Gaussian Orthogonal Ensemble of random matrices with a potential function of degree d. Our Riemann-Hilbert problem is similar to a local d × d RiemannHilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polyno...

We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large N matrix model. Using a related function Ξ(z) we develop an analog between this function and the Airy function Ai(z) of the Gaussian matrix model. The analog...

We compare MC calculations of the density of states in SU(2) pure gauge theory with the weak and strong coupling expansions. Surprisingly, the range of validity of the two approximations overlap significantly, however the large order behavior of both expansions appear to be similar to the corresponding expansions of the plaquette. We discuss the implications for the calculation of the Fisher’s ...

In this article, I present a simple conjecture on the number of independent sets on graph covers. The conjecture implies that the partition function of a binary pairwise attractive model is greater than that of the Bethe approximation.

We study right tail large deviations of the logarithm of the partition function for directed lattice paths in i.i.d. random potentials. The main purpose is the derivation of explicit formulas for the 1+1-dimensional exactly solvable case with log-gamma distributed random weights. Along the way we establish some regularity results for this rate function for general distributions in arbitrary dim...