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

TOMORROW

#### Faculty Teaching and Learning Showcase/CARMA Seminar

"Creating a high expectations learning context for university level mathematics teaching"
Dr David Collis

4:00 pm, Thu, 16th Aug 2018
L326, Auchmuty Library

#### Zero-Dimensional Symmetry Seminar

"Locally pro-p contraction groups are nilpotent"
Prof George Willis

2:00 pm, Mon, 20th Aug 2018
G29, McMullin

WORKSHOP

Sat, 25th Aug 2018 — Sun, 26th Aug 2018
Harbourview Function Centre (Newcastle, NSW)

The first ever AVOCADO (Analysis of Variations, Optimal Control, and Applications to Design and Operations) workshop will be held in Newcastle on August 25th and 26th and is hosted by CARMA. Details can be found on the workshop webpage.

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

# NEWS

#### Statistical Society of Australia service award for Peter Howley

Congratulations to A/Prof Peter Howley who has received word from the President of the Statistical Society of Australia (SSA) advising that he has been awarded the 2018 SSA Service Awar... [READ MORE]

#### Dr Minh Dao wins best poster at AMSI Optimise

CARMA member Dr Minh Dao was the winner of the AMSI Optimise 2018 Best Poster Competition for his poster entitled "Optimisation Design for Energy-Efficient Downlink Cloud Radio Access N... [READ MORE]

#### "Theory and simulation in physics for materials applications" symposium

CARMA member Elena Levchenko is a co-organising for the symposium "Theory and simulation in physics for materials applications" at this year's European Materials Research Society (E-MRS... [READ MORE]

# HIGHLIGHTS

Selected paper from DocServer
Jonathan M. Borwein, Stephen Choi

ABSTRACT

In \cite{HW}, Hardy and Wright recorded elegant closed forms for the generating functions of the divisor functions $\sigma_k(n)$ and $\sigma_k(n)^2$:$\sum_{n=1}^\infty \frac{\sigma_k(n)}{n^s}=\zeta (s)\zeta (s-k)$ and $\sum_{n=1}^\infty \frac{\sigma_k(n)^2}{n^s}=\frac{\zeta (s)\zeta (s-k)^2\zeta (s-2k)}{\zeta (2s-2k)}.$ In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem \ref{thm 2.1} below, we are able to generalize the above result and prove that if $f_i$ and $g_i$ are completely multiplicative, then we have$\sum_{n=1}^\infty \frac{(f_1\ast g_1)(n)\cdot (f_2\ast g_2)(n)}{n^s}=\frac{L_{f_1f_2}(s)L_{g_1g_2}(s)L_{f_1g_2}(s)L_{g_1f_2}(s)}{L_{f_1f_2g_1g_2}(2s)}$ where $L_f(s):=\sum_{n=1}^\infty f(n)n^{-s}$ is the Dirichlet series corresponding to $f$. Let $r_N(n)$ be the number of solutions of $x_1^2+\cdots +x_N^2=n$ and $r_{2,P}(n)$ be the number of solutions of $x^2+Py^2=n$. One of the applications of Theorem \ref{thm 2.1} is to obtain closed forms, in terms of $\zeta (s)$ and Dirichlet $L$-functions, for the generating functions of $r_N(n), r_N(n)^2, r_{2,P}(n)$ and $r_{2,P}(n)^2$. We also use these generating functions to obtain asymptotic estimates of the average values for all these functions.

# 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.