Shane Chern

activity

20162022

most citedOn the power mean of a sum analogous to the Kloosterman sum

10 citations · 11 across the 11 of their papers we have counts for

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14 papers · 1 filter

math.NT2022

Asymmetric Rogers--Ramanujan type identities. I. The Andrews--Uncu Conjecture

Shane Chern

In this work, we start an investigation of asymmetric Rogers--Ramanujan type identities. The first object is the following unexpected relation $$\sum_{n\ge 0} \frac{(-1)^n q^{3\bin…

math.NT2021

On a congruence involving harmonic series and Bernoulli numbers

Shane Chern

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime $p$ , $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,p)=1\ı+j+k=p}}\frac{1}{…

math.NT2019

Asymptotics for the Taylor coefficients of certain infinite products

Shane Chern

Let $(m_{1}, \dots, m_{J})$ and $(r_{1}, \dots, r_{J})$ be two sequences of $J$ positive integers satisfying $1 \leq r_{j} < m_{j}$ for all $j = 1, \dots, J$ . Let $(δ_{1}, \dots, δ_{J})$ be a sequence of $…

math.NT2018

Note on sums involving the Euler function

Shane Chern

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} ϕ\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qqu…

math.NT2018

Linked partition ideals and Kanade--Russell conjectures

Shane Chern, Zhitai Li

This paper will primarily present a method of proving generating function identities for partitions from linked partition ideals. The method we introduce is built on a conjecture b…

math.NT2018

A further look at the truncated pentagonal number theorem

Shane Chern

In this paper, we study the asymptotic behavior of the following function $M_{k} (n) := (- 1)^{k - 1} j = 0 \sum k - 1 (p (n - j (3 j + 1) /2) - p (n - j (3 j + 5) /2 - 1)),$ which arises from Andrew…