arXiv subjects

Renrong Mao

Publications and source records attributed to Renrong Mao.

7 recordsLinked to original sources

arXiv2026

On the modularity of the odd rank generating functions

To provide partition-theoretic interpretations to Watson's the third-order mock theta function $ω(q)$, Andrews defined the odd Durfee symbols and odd ranks. Motivated by Andrews' work, arithmetic properties of odd ranks are widely studied recently. In this paper, we obtain transformation formulas of the odd rank generating functions, which are used to construct families of weak Maass forms and weakly modular forms. As an application,we provide explicit identities for odd ranks modulo 5, analogous to Ramanujan's classical partition identities.

math.NT↗

arXiv2024

Congruences modulo powers of $5$ for odd ranks

In 2007, Andrews studied the odd Durfee symbols and their odd ranks. Let $N^0(m,k,n)$ denote the number of odd Durfee symbols of $n$ with odd rank congruent to $m$ modulo $k$. Motivated by Andrews' work, many authors obtained generating functions of $N^0(m,k,n)$ from which relations between odd ranks are proved. In this paper, we establish a family of congruences for odd ranks modulo powers of $5$.

math.CO↗

arXiv2024

Asymptotic formulas for the coefficients of the truncated theta series

Motivated by the groundbreaking work of Andrews and Merca, truncated theta series have been extensively studied over the years. In particular, Merca made conjectures on the non-negativity of the coefficient of $q^N$ in truncated series from the Jacobi triple product identity and the quintuple product identity. In this paper, using Wright's Circle Method, we establish asymptotic formulas for the coefficients of truncated theta series and prove that Merca's conjectures are true for sufficiently large $N$.

math.CO↗

arXiv2023

A proof of a conjecture of Mao on Beck's partition statistics modulo 8

Beck introduced two partition statistics $NT(r,m,n)$ and $M_ω(r,m,n)$,which denote the total number of parts in the partition of $n$ with rank congruent to $r$ modulo $m$ and the total number of ones in the partition of $n$ with crank congruent to $r$ modulo $m$, respectively. In recent years, a number of congruences and identities on $NT(r,m,n)$ and $M_ω(r,m,n)$ for some small $m $ have been established.In this paper, we prove an identity on $NT(r,8,n)$ and $M_ω(r,4,n)$ which confirm a conjecture given by Mao.

math.NT↗

arXiv2023

Andrews-Beck type congrences modulo powers of 5

Let $NT(m, k, n)$ denote the total number of parts in the partitions of n with rank congruent to m modulo k. Andrews proved Beck's conjecture on congruences for $NT(m, k, n)$ modulo 5 and 7. Generalizing Andrews'results, Chern obtain congruences for $NT(m, k, n)$ modulo 11 and 13. More recently, the second author use the theory of Hecke operators to establish congruences for such partition statistics modulo powers of primes $\ell \ge 7$. In this paper, we obtain Andrews-Beck type congruences modulo powers of 5.

math.NT↗

arXiv2020

Variations of Andrews-Beck type congruences

We prove three variations of recent results due to Andrews on congruences for $NT(m,k,n)$, the total number of parts in the partitions of $n$ with rank congruent to $m$ modulo $k$. We also conjecture new congruences and relations for $NT(m,k,n)$ and for a related crank-type function.

math.NT↗

arXiv2015

On recursions for coefficients of mock theta functions

We use a generalized Lambert series identity due to the first author to present q-series proofs of recent results of Imamoglu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order mock theta functions. Additionally, we discuss an application of this identity to other mock theta functions.

math.NT↗