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 before transitioning into his current role as Head of Academic Partnerships. He spends most of his time talking to students about all of the exciting applications of mathematics in the real world (and quite often he talks about prime numbers).
Research Interests
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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: 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
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Journal Article: 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
Available Projects
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The Distribution of Prime Numbers
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.
Another problem involves the additive role that primes play. The Goldbach conjecture is the assertion that any even number greater than two can be expressed as the sum of two primes. There are many related similiar results that one can prove armed with the tools of modern number theory.
Publications
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
-
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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The Distribution of Prime Numbers
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.
Another problem involves the additive role that primes play. The Goldbach conjecture is the assertion that any even number greater than two can be expressed as the sum of two primes. There are many related similiar results that one can prove armed with the tools of modern number theory.
