CARMA supports the formation of the National Research Centre by the Australian Mathematical Sciences Institute. CARMA will be a foundation partner in the Centre when it is launched in 2016.

# RESEARCH

CARMA renewed until 2020: read our presentation here

# EVENTS

#### CARMA Seminar

"The complexity of the isomorphism problem for t.d.l.c. and other types of groups"
Prof. André Nies

4:00 pm, Thu, 1st Jun 2017
V205, Mathematics Building

These are the events in the next 7 days. For more, see the events page.

# NEWS

#### CARMA RHD Students Win Prizes for Mathematical Visualisations

The postgraduate students in the School of Mathematical and Physical Sciences celebrated their research through an artistic exhibition curated by Dr Michael Gladys. First prize went to ... [READ MORE]

#### 2014 Euler Medal awarded to CARMA member Brian Alspach

Brian Alspach has been one of the most influential graph theorists over the last five decades. He has contributed pioneering works, fundamental discoveries, and celebrated results in ar... [READ MORE]

#### Bishnu Lamichhane now Chair of the Computational Mathematics Group Aus

Bishnu has been elected as the new chair of ANZIAM's Computational Mathematics Group.

# HIGHLIGHTS

Selected paper from DocServer
Jonathan M. Borwein, Stephen Choi

ABSTRACT

Recently, Crandall in \cite{Cr} used Andrews' identity for the cube of the Jacobian theta function $\theta_4$: $\theta_4^3(q)=\left(\sum_{n \in \IZ}(-1)^nq^{n^2}\right)^3=1+4\sum_{n=1}^{\infty}\frac{(-1)^nq^n}{1+q^n}-2\sum_{\substack{n=1 \\ |j| < n}}^{\infty}\frac{q^{n^2-j^2}(1-q^n)(-1)^j}{1+q^n}$ to derive new representations for Madelung's constant and various of its analytic relatives. He considered the three-dimensional Epstein zeta function $M(s)$ which is the analytic continuation of the series $\sum_{\substack{x,y,z\in \IZ \\ (x,y,z)\neq (0,0,0)}}\frac{(-1)^{x+y+z}}{(x^2+y^2+z^2)^{s}}.$ Then the number $M(\frac{1}{2})$ is the celebrated {\em Madelung constant}. Using a reworking of the above mentioned Andrews' identity, he obtained the formula $M(s)=-6(1-2^{1-s})^2 \zeta^2(s)-4U(s)$ where $\zeta (s)$ is the Riemann zeta function and $U(s):=\sum_{x,y,z \ge 1}\frac{(-1)^{x+y+z}}{(xy+yz+xz)^s}.$ In view of this representation, Crandall asked what integers are of the form of $xy+yz+xz$ with $x,y,z \ge 1$ and he made a conjecture that every odd integer greater than one can be written as $xy+yz+xz$. In this manuscript, we shall show that Crandall's conjecture is indeed true.

# MEMBERSHIP

Membership to CARMA offers many benefits and is available by invitation to all University of Newcastle academic staff. Associate membership, also by invitation, is available to external researchers and practitioners for three-year renewable terms. Associate members are expected to visit CARMA with some frequency, typically for a total of three to four weeks in a year, and to be involved in one or more ongoing research projects with CARMA members. CARMA is able to assist with the travel and living costs of such visits.