Bulletin of the Australian Mathematical Society

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2021 highlights

Contents

ADDITIVE BASES AND NIVEN NUMBERS
Part of:
- Elementary number theory
- Sequences and sets
CARLO SANNA
Journal: Bulletin of the Australian Mathematical Society / Volume 104 / Issue 3 / 2021
Published online by Cambridge University Press: 25 March 2021, pp. 373-380
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Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

ON A WEIGHTED SUM OF MULTIPLE $\mathbf{{T}}$ -VALUES OF FIXED WEIGHT AND DEPTH
Part of:
- Zeta and $L$-functions: analytic theory
YOSHIHIRO TAKEYAMA
Journal: Bulletin of the Australian Mathematical Society / Volume 104 / Issue 3 / December 2021
Published online by Cambridge University Press: 19 March 2021, pp. 398-405
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The multiple T-value, which is a variant of the multiple zeta value of level two, was introduced by Kaneko and Tsumura [‘Zeta functions connecting multiple zeta values and poly-Bernoulli numbers’, in: Various Aspects of Multiple Zeta Functions, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204]. We show that the generating function of a weighted sum of multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type.

CONGRUENCES MODULO 4 FOR WEIGHT $\textbf{3/2}$ ETA-PRODUCTS
Part of:
- Forms and linear algebraic groups
- Discontinuous groups and automorphic forms
RONG CHEN, F. G. GARVAN
Journal: Bulletin of the Australian Mathematical Society / Volume 103 / Issue 3 / June 2021
Published online by Cambridge University Press: 05 October 2020, pp. 405-417
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We find and prove a class of congruences modulo 4 for eta-products associated with certain ternary quadratic forms. This study was inspired by similar conjectured congruences modulo 4 for certain mock theta functions.

POINTS OF SMALL HEIGHT ON AFFINE VARIETIES DEFINED OVER FUNCTION FIELDS OF FINITE TRANSCENDENCE DEGREE
Part of:
- Arithmetic problems. Diophantine geometry
- Arithmetic algebraic geometry
DRAGOS GHIOCA, DAC-NHAN-TAM NGUYEN
Journal: Bulletin of the Australian Mathematical Society / Volume 103 / Issue 3 / June 2021
Published online by Cambridge University Press: 05 October 2020, pp. 418-427
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We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$ . Furthermore, we obtain sharp lower bounds for the Weil height of the points in $V(\overline {K})$ , which are not contained in the largest subvariety $W\subseteq V$ defined over the constant field $\overline {k}$ .

FUSION 2-CATEGORIES WITH NO LINE OPERATORS ARE GROUPLIKE
THEO JOHNSON-FREYD, MATTHEW YU
Journal: Bulletin of the Australian Mathematical Society / Volume 104 / Issue 3 / December 2021
Published online by Cambridge University Press: 19 February 2021, pp. 434-442
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We show that if ${\mathcal C}$ is a fusion $2$ -category in which the endomorphism category of the unit object is or , then the indecomposable objects of ${\mathcal C}$ form a finite group.

A VARIANT OF CAUCHY’S ARGUMENT PRINCIPLE FOR ANALYTIC FUNCTIONS WHICH APPLIES TO CURVES CONTAINING ZEROS
Part of:
- Miscellaneous topics of analysis in the complex domain
MAHER BOUDABRA, GREG MARKOWSKY
Journal: Bulletin of the Australian Mathematical Society / Volume 103 / Issue 3 / June 2021
Published online by Cambridge University Press: 18 January 2021, pp. 486-492
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The standard version of Cauchy’s argument principle, applied to a holomorphic function f, requires that f has no zeros on the curve of integration. In this note, we give a generalisation of such a principle which covers the case when f has zeros on the curve, as well as an application.