K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products

K. Saito's Conjecture for Nonnegative Eta Products

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z = 1 case is an identity for the generating function for p-cores due to Klyachko [12] and Garvan, Kim and Stanton [7].

从同伦角度看的分歧理论，I

Ramification theory from homotopical point of view, I

本文从同伦角度研究分歧理论，证明了proper映射下 étale sheaf 的推进与特征环推进在p-torsion下的兼容性，采用∞-范畴理论实现。

New Identities for 7-cores with prescribed BG-rank

A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of partitions of n that are 7-cores with given BG-rank can be written as certain sum of multi-theta functions. We give explicit representations for these generating functions in terms of sums of positive eta-quotients and derive inequalities for the their coefficients. New identities for the generating function of unrestricted 7-cores and inequalities for their coe…

傅里叶求和公式与晶体测度的分类

A classification of Fourier summation formulas and crystalline measures

本文对傅里叶求和公式和晶体测度进行了分类，并引入了eta-quotients方法推广了Guinand的例子。

Arithmetic properties of certain $t$-regular partitions

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of $b_9(n)$ and $b_{19}(n)$ modulo $2$. We also relate $b_{t}(n)$ to the ordinary partition function, and prove that $b_{t}(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. For $t\in \{6,10,14,15,18,20,22,2…