# Recent Preprints and Publications

These are all in PDF format. For full details on the publication status of any item in the list
below contact Kevin Broughan at
kab@waikato.ac.nz

The holomorphic flow of the Riemann zeta function,
with A. R. Barnett, (Mathematics of Computation, vol 73 (2004), p987-1004.)

Corrigendum for "The holomorphic flow of the Riemann zeta function",
with A. R. Barnett, Math. Comp. vol 76 (2007), p2249-2250.

Corrigendum II for "The holomorphic flow of the Riemann zeta function",
with A. R. Barnett, (preprint).

Phase portraits of the zeros of the Riemann Zeta function,
(web data base).

Phase portraits of the zeros of the Riemann Xi function,
(web data base).

Holomorphic flows on simply connected regions have no limit cycles,
Meccanica, vol 38 (2003), p699-709.

Corrigenda for "Holomorphic flows on simply connected regions have no limit cycles", Meccanica, vol 42 (2007) p213.

The gcd-sum function,
Journal of Integer Sequences, vol 4 (2002) Article 01.2.2, p1-19.

Restricted divisor sums,
Acta Arithmetica, vol 101, (2002), p105-114.

Relationships between the integer conductor and k'th root functions,
International Journal of Pure and Applied Mathematics, vol 5 (2003), p253-275.

Vanishing of the integral of the Hurwitz zeta function,
Bulletin of the Australian Mathematical Society, vol 65 (2002) p121-127.

Adic topologies for the rational integers,,
Canadian Journal of Mathematics, vol 55 (2003), p711-723.

Characterizing the sum of two cubes,
Journal of Integer Sequences, vol 6 (2003), Article 03.4.6, p1-7.

Asymptotic order of the square free part of n!,
Integers: The Electronic Journal of Combinatorial Number Theory, vol 2, A10, (2002), p1-6.

Relaxations of the ABC conjecture using integer k'th
roots, New Zealand Journal of Mathematics, vol 35 (2006), p121-136.

The structure of sectors of zeros of entire flows
Topology Proceedings, vol 27 (2003), p379-394.

Holomorphic flow of Riemann's function Xi(s)
Nonlinearity, vol 18, (2005), p1269-1294.

Linear law for the logarithms of the Riemann periods at simple critical zeta zeros,
with A. R. Barnett, Mathematics of Computation, vol 75 (2006), p891-902.

Extension of the Riemann xi-function's logarithmic derivative positivity
region to near the critical strip, Canadian Mathematical Bulletin, vol 52 (2009), p186-194.

Harmonic sets and the harmonic prime number theorem, with R. J. Casey,
Bulletin of the Australian Math. Soc. vol 71, (2005), p127-137.

The average order of the Dirichlet series of the gcd-sum function,
Journal of Integer Sequences, vol 10, (2007), Article 07.4.2, p1-6.

Odd multiperfect numbers of abundancy four, with Qizhi Zhou, Journal of Number Theory, vol 128 (2008), p1568-1575.

Gram lines and the average of the real part of the Riemann
zeta function, with A. Ross Barnett, electronically published 7th Dec 2011, Math. Comp. vol 81 (2012), 1669-1679.

A note on Casimir operators for the generalized upper half plane, International Journal of Pure and Applied Mathematics, 41 (2007), p549-560.

Evaluating Jacquet's GL(n) Whittaker function, Mathematics of Computation, 78 (2009), p1061-1072.

On primes with prime subscripts, with A. Ross Barnett, J. of Integer Sequences, 12 (2009) Article 09.2.3, p1-12.

Twelfth power residue difference sets, with K. Byard, Bull. Aust. Math. Soc., vol 80 (2009), 147-158.

An algorithm for the explicit evaluation of GL(n) Kloosterman sums, ACM Communications in Computer Algebra, vol 43 (2009), p1-10.

On the number of missing values of n! modulo p, with A. Ross Barnett, Journal of the Ramanujan Mathematical Society, vol 24 (2009). p277-284.

On flat primes and thin primes, with Zhou Qizhi, Bull. Aust. Math. Soc. vol 82 (2010), p282-292.

Some Divisibility Properties of Binomial Coefficients and Wolstenholme's conjecture, with Florian Luca and Igor Shparlinski, Integers, vol 10 (2010), p485-495.

On the Furstenberg closure of a class of binary recurrences, with Florian Luca, J. Number Theory, vol 130 (2010), p696-706.

There are No Multiply-Perfect Fibonacci Numbers,
with Marcos J. Gonzalez, Ryan H. Lewis,
Florian Luca, V. Janitzio Mejia Huguet and Alain Togbe, Integers vol 10 (2010), 363-397.

Perfect Repdigits, with Florian Luca and Sergio Guzman Sanchez, Math. Comp. vol 82 (2013), 2439-2459.

Odd repdigits to small bases are not perfect, with Qizhi Zhou, Integers, vol 12 (2012), 787-1089.

An explicit bound for aliquot cycles of repdigits, Integers, vol 12 (2012), 689-704.

Sum-products estimates with several sets and applications, with A. Balog and I. Shparlinski, Integers, (2012), 895-906.

Quadrafree factorisatio numerorum, Rocky Mountain J. Math., vol 44 (2014), 791-807.

On shifted primes and balanced primes, International Journal of
Number Theory, vol 8 (2012), 2017-2035.

On integers for which the sum of divisors is the square of the
squarefree core, with J.M. De Koninck, I. Katai and F. Luca, Journal of Integer Squences, vol 15
(2012) article 12.7.5, 1-12.

On square values of the product of the Euler totient function and sum
of divisors function, with K. Ford and F. Luca, Colloquium Mathematicum, vol 130 (2013), 127-137.

Improving the Chen and Chen result for odd perfect numbers, with Daniel Delbourgo, Integers vol 13 (2013) Article A-39, 1-8.

On the ratio of the sum of divisors and Euler's totient function I, with Daniel Delbourgo, Journal of Integer Sequences, vol 16 (2013) Art 13.8.8, 1-16.

On the ratio of the sum of divisors and Euler's totient function II, with Qizhi Zhou, Journal of Integer Sequences, vol 17 (2014) Art 14.9.2., 1-22.

A conjecture of De Koninck regarding particular square values of the sum of divisors function, with Daniel Delbourgo and Qizhi Zhou, Journal of Number Theory, vol 137 (2014), 50-66.

Robin's inequality for 11-free integers, with Tim Trudgian, Integers, vol 15 (2015), 1-5.

*Last modified: 10th April 2016.*