Adjunct Associate Professor
Overview
Adrian grew up in Perth and double majored in Pure Mathematics and Applied Mathematics at the University of Western Australia. Soonafter, he ventured to Canberra to undertake a PhD, focussing on analytic number theory: an enchanting area where one perplexingly uses calculus and analysis to study discrete structures such as the set of prime numbers.
After this, he worked as a derivatives trader at Optiver APAC for five years and stayed on there as Head of Academic Partnerships. He currently straddles both industry and academia and believes they both have much to offer mathematicians.
Adrian is available (and invariably keen) to supervise honours, masters and PhD projects in analytic number theory.
Research Interests
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Analytic Number Theory
Analytic Number Theory is the application of tools from analysis to study the structure of discrete sets such as the prime numbers. This is a beautiful field with much interplay between the continuous and discrete. -
The Distribution of Prime Numbers
The prime numbers are simple to define, yet try to study them in depth and they are notoriously evasive. I enjoy all manner of problems involving prime numbers, from results that establish the existence of prime numbers in short intervals to additive problems similiar to Goldbach's conjecture. -
Construction of Expander Graphs
An expander graph is a graph that is a strong candidate to be used as some sort of network. Mathematicians have a straightfoward way of characterising the expansion of a graph, that is, how well information travels through it. It then becomes an interesting problem to construct arbitrarily large graphs with high expansion. Solutions to this problem call on various seemingly disconnected areas of maths.
Publications
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Journal Article: An explicit mean-value estimate for the prime number theorem in interval
Cully-Hugill, Michaela and Dudek, Adrian W. (2023). An explicit mean-value estimate for the prime number theorem in interval. Journal of the Australian Mathematical Society, 1-15. doi: 10.1017/S1446788723000113
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Journal Article: A conditional explicit result for the prime number theorem in short intervals
Cully-Hugill, Michaela and Dudek, Adrian W. (2022). A conditional explicit result for the prime number theorem in short intervals. Research in Number Theory, 8 (3) 61. doi: 10.1007/s40993-022-00358-1
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Journal Article: Explicit short intervals for primes in arithmetic progressions on GRH
Dudek, Adrian W., Grenie, Loic and Molteni, Giuseppe (2019). Explicit short intervals for primes in arithmetic progressions on GRH. International Journal of Number Theory, 15 (4), 825-862. doi: 10.1142/s1793042119500441
Available Projects
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Prime Numbers in Short Intervals (PhD, Masters, Honours)
The Riemann Hypothesis is possibly the most well known unsolved problem in all of mathematics. Incredibly, the more we know about the zeroes of the Riemann zeta-function, the more we know about the set of prime numbers. There are entire books full of unsolved problems on prime numbers; recent advances in studying zeta-function zeroes can be called upon to tackle these.
One such problem involves proving the existence of a function h(x) such that the interval (x, x+h(x)) always contains at least one prime. The goal of this area of research is to find slow-growing functions that do the trick; this would ensure the existence of primes in quite small intervals. Much of this research is motivated by Legendre's conjecture: the unproved assertion that there is always a prime number between any two square numbers.
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Problems in Additive Number Theory (PhD, Masters, Honours)
Goldbach's conjecture is the assertion that every even integer greater than 2 can be written as the sum of two prime numbers. Despite enormous efforts, this has still not been proven and remains out of reach.
There are plenty of problems of the above form that one can prove using a variety of techniques. In this project, the candidate will work on tangible problems in additive number theory using classic analytic approaches such as the distribution of prime numbers and sieve methods.
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Applications of Number Theory to Graph Theory and Group Theory (PhD, Masters, Honours)
Many problems in algebra reduce to establishing results in number theory. For example, the construction of expander graphs or estimating the number of non-isomorphic groups of a given order can be tackled using number theoretic tools.
In this project, one will work at the intersection, tackling problems of an algebraic birthplace with tools and results from (analytic) number theory.
Publications
Journal Article
-
An explicit mean-value estimate for the prime number theorem in interval
Cully-Hugill, Michaela and Dudek, Adrian W. (2023). An explicit mean-value estimate for the prime number theorem in interval. Journal of the Australian Mathematical Society, 1-15. doi: 10.1017/S1446788723000113
-
A conditional explicit result for the prime number theorem in short intervals
Cully-Hugill, Michaela and Dudek, Adrian W. (2022). A conditional explicit result for the prime number theorem in short intervals. Research in Number Theory, 8 (3) 61. doi: 10.1007/s40993-022-00358-1
-
Explicit short intervals for primes in arithmetic progressions on GRH
Dudek, Adrian W., Grenie, Loic and Molteni, Giuseppe (2019). Explicit short intervals for primes in arithmetic progressions on GRH. International Journal of Number Theory, 15 (4), 825-862. doi: 10.1142/s1793042119500441
-
Note on the number of divisors of reducible quadratic polynomials
Dudek, Adrian W., Pankowski, Lukasz and Scharaschkin, Victor (2019). Note on the number of divisors of reducible quadratic polynomials. Bulletin of the Australian Mathematical Society, 99 (1), 1-9. doi: 10.1017/S0004972718000734
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On the sum of a prime and a square-free number
Dudek, Adrian W. (2017). On the sum of a prime and a square-free number. Ramanujan Journal, 42 (1), 233-240. doi: 10.1007/s11139-015-9736-2
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An explicit result for primes between cubes
Dudek, Adrian W. (2016). An explicit result for primes between cubes. Functiones Et Approximatio Commentarii Mathematici, 55 (2), 177-197. doi: 10.7169/facm/2016.55.2.3
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On the Success of Mishandling Euclid's Lemma
Dudek, Adrian W. (2016). On the Success of Mishandling Euclid's Lemma. American Mathematical Monthly, 123 (9), 924-927. doi: 10.4169/amer.math.monthly.123.9.924
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On the Spectrum of the Generalised Petersen Graphs
Dudek, Adrian W. (2016). On the Spectrum of the Generalised Petersen Graphs. Graphs and Combinatorics, 32 (5), 1843-1850. doi: 10.1007/s00373-016-1676-0
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Primes in explicit short intervals on RH
Dudek, Adrian W., Grenie, Loic and Molteni, Giuseppe (2016). Primes in explicit short intervals on RH. International Journal of Number Theory, 12 (5), 1391-1407. doi: 10.1142/s1793042116500858
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On the number of divisors of n2−1
Dudek, Adrian W. (2016). On the number of divisors of n2−1. Bulletin of the Australian Mathematical Society, 93 (2), 194-198. doi: 10.1017/s0004972715001136
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Almost-Ramanujan graphs and prime gaps (vol 43, pg 204, 2015)
Dudek, Adrian W. (2016). Almost-Ramanujan graphs and prime gaps (vol 43, pg 204, 2015). European Journal of Combinatorics, 51, 533-534. doi: 10.1016/j.ejc.2015.06.002
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On the sum of the square of a prime and a square-free number
Dudek, Adrian W. and Platt, David J. (2016). On the sum of the square of a prime and a square-free number. LMS Journal of Computation and Mathematics, 19 (1), 16-24. doi: 10.1112/s1461157015000297
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An explicit result for |L(1+it,χ)|
Dudek, Adrian W. (2015). An explicit result for |L(1+it,χ)|. Functiones Et Approximatio Commentarii Mathematici, 53 (1), 23-29. doi: 10.7169/facm/2015.53.1.2
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On the Riemann hypothesis and the difference between primes
Dudek, Adrian W. (2015). On the Riemann hypothesis and the difference between primes. International Journal of Number Theory, 11 (3), 771-778. doi: 10.1142/s1793042115500426
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Almost-Ramanujan graphs and prime gaps
Dudek, Adrian W. (2015). Almost-Ramanujan graphs and prime gaps. European Journal of Combinatorics, 43, 204-209. doi: 10.1016/j.ejc.2014.09.001
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On Solving a Curious Inequality of Ramanujan
Dudek, Adrian W. and Platt, David J. (2015). On Solving a Curious Inequality of Ramanujan. Experimental Mathematics, 24 (3), 289-294. doi: 10.1080/10586458.2014.990118
Possible Research Projects
Note for students: The possible research projects listed on this page may not be comprehensive or up to date. Always feel free to contact the staff for more information, and also with your own research ideas.
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Prime Numbers in Short Intervals (PhD, Masters, Honours)
The Riemann Hypothesis is possibly the most well known unsolved problem in all of mathematics. Incredibly, the more we know about the zeroes of the Riemann zeta-function, the more we know about the set of prime numbers. There are entire books full of unsolved problems on prime numbers; recent advances in studying zeta-function zeroes can be called upon to tackle these.
One such problem involves proving the existence of a function h(x) such that the interval (x, x+h(x)) always contains at least one prime. The goal of this area of research is to find slow-growing functions that do the trick; this would ensure the existence of primes in quite small intervals. Much of this research is motivated by Legendre's conjecture: the unproved assertion that there is always a prime number between any two square numbers.
-
Problems in Additive Number Theory (PhD, Masters, Honours)
Goldbach's conjecture is the assertion that every even integer greater than 2 can be written as the sum of two prime numbers. Despite enormous efforts, this has still not been proven and remains out of reach.
There are plenty of problems of the above form that one can prove using a variety of techniques. In this project, the candidate will work on tangible problems in additive number theory using classic analytic approaches such as the distribution of prime numbers and sieve methods.
-
Applications of Number Theory to Graph Theory and Group Theory (PhD, Masters, Honours)
Many problems in algebra reduce to establishing results in number theory. For example, the construction of expander graphs or estimating the number of non-isomorphic groups of a given order can be tackled using number theoretic tools.
In this project, one will work at the intersection, tackling problems of an algebraic birthplace with tools and results from (analytic) number theory.
-
Bounded Gaps Between Prime Numbers (Honours)
The Twin Prime Conjecture states that there are infinitely many pairs of primes that are two apart. This remains unproven but astounding progress was made in 2013 when Yitang Zhang proved that there exists some finite number such that there are infinitely many primes with this number as their difference.
In this project, students will learn some analytic number theory towards understanding how such a result can be proven. They will also be able to highlight where progress is expected to be made and also work on some related but more tractable problems.
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Fourier Analysis in Number Theory (Honours)
It is well known (see, for example, the Riemann Hypothesis) that there is an intimate connection between the zeroes of the Riemann zeta-function and the distribution of prime numbers. This can be seen more directly in the setting of Fourier transforms; indeed, many modern proofs of the Prime Number Theorem opt for this approach. In this project, we will work to understand this connection more deeply as well as apply it towards new results in analytic number theory.
