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Professor Timothy Trudgian

Professor Timothy Trudgian

Professor
UNSW Canberra
School of Science

BSc (Hons), The Australian National University, December 2005

DPhil, University of Oxford, June 2010

Research Interests: analytic number theory: the Riemann zeta-function, distribution of primes, primitive roots.

My CV; see also my UNSW Canberra website

I am part of the Number Theory Group at UNSW Canberra; I work closely with colleagues in number theory at UNSW Sydney.

Scholarships of $37,500 (AUD) are available for PhD applicants who achieved H1 (High Distinction) in their undergradu...
Phone
+61 2 5114 5026
E-mail
t.trudgian@adfa.edu.au
Location
Room 104 PEMS South (Bldg 26) School of Physical, Environmental and Mathematical Sciences, UNSW Canberra, Australian Defence Force Academy, PO Box 7916, Canberra BC ACT 2610, AUSTRALIA
  • Book Chapters | 2020
    Mossinghoff M; Trudgian T, 2020, 'A tale of two omegas', in , American Mathematical Society, pp. 343 - 364, http://dx.doi.org/10.1090/conm/754/15149
    Book Chapters | 2017
    Mossinghoff M; Trudgian T, 2017, 'The Liouville function and the Riemann hypothesis', in Montgomery H; Nikeghbali A; Rassias M (ed.), Exploring the Riemann Zeta Function 190 years from Riemann's Birth, Springer, Cham, pp. 201 - 221
  • Journal articles | 2025
    TRUDGIAN TS; YANG A, 2025, 'TOWARD OPTIMAL EXPONENT PAIRS', Mathematics of Computation, 94, pp. 1467 - 1502, http://dx.doi.org/10.1090/mcom/3993
    Journal articles | 2024
    Mossinghoff MJ; Trudgian TS; Yang A, 2024, 'Explicit zero-free regions for the Riemann zeta-function', Research in Number Theory, 10, http://dx.doi.org/10.1007/s40993-023-00498-y
    Journal articles | 2023
    Bellotti C; Keller TM; Trudgian TS, 2023, 'New bounds for numbers of primes in element orders of finite groups', Mathematische Nachrichten, 296, pp. 5227 - 5231, http://dx.doi.org/10.1002/mana.202200484
    Journal articles | 2023
    Goldston DA; Trudgian TS; Turnage-Butterbaugh CL, 2023, 'On the Montgomery–Odlyzko method regarding gaps between zeros of the zeta-function', Journal of Mathematical Analysis and Applications, 527, http://dx.doi.org/10.1016/j.jmaa.2023.127548
    Journal articles | 2023
    Johnston DR; Ramaré O; Trudgian T, 2023, 'An explicit upper bound for L(1 , χ) when χ is quadratic', Research in Number Theory, 9, http://dx.doi.org/10.1007/s40993-023-00476-4
    Journal articles | 2023
    Martin G; Mossinghoff MJ; Trudgian TS, 2023, 'Fake mu’s', Proceedings of the American Mathematical Society, 151, pp. 3229 - 3244, http://dx.doi.org/10.1090/proc/16186
    Journal articles | 2022
    Booker AR; Cohen SD; Leong N; Trudgian T, 2022, 'PRIMITIVE ELEMENT PAIRS with A PRESCRIBED TRACE in the CUBIC EXTENSION of A FINITE FIELD', Bulletin of the Australian Mathematical Society, 106, pp. 458 - 462, http://dx.doi.org/10.1017/S0004972722000442
    Journal articles | 2022
    Booker AR; Cohen SD; Leong N; Trudgian T, 2022, 'Primitive elements with prescribed traces', Finite Fields and their Applications, 84, http://dx.doi.org/10.1016/j.ffa.2022.102094
    Journal articles | 2022
    Booker AR; Hathi S; Mossinghoff MJ; Trudgian TS, 2022, 'Wolstenholme and Vandiver primes', Ramanujan Journal, 58, pp. 913 - 941, http://dx.doi.org/10.1007/s11139-021-00438-3
    Journal articles | 2022
    Brent RP; Platt DJ; Trudgian TS, 2022, 'The mean square of the error term in the prime number theorem', Journal of Number Theory, 238, pp. 740 - 762, http://dx.doi.org/10.1016/j.jnt.2021.09.016
    Journal articles | 2022
    Jarso T; Trudgian T, 2022, 'FOUR CONSECUTIVE PRIMITIVE ELEMENTS IN A FINITE FIELD', Mathematics of Computation, 91, pp. 1521 - 1532, http://dx.doi.org/10.1090/mcom/3716
    Journal articles | 2022
    Languasco A; Trudgian TS, 2022, 'Uniform effective estimates for', Journal of Number Theory, 236, pp. 245 - 260, http://dx.doi.org/10.1016/j.jnt.2021.07.019
    Journal articles | 2022
    Morrill T; Platt D; Trudgian T, 2022, 'Sign changes in the prime number theorem', Ramanujan Journal, 57, pp. 165 - 173, http://dx.doi.org/10.1007/s11139-021-00398-8
    Journal articles | 2022
    Mossinghoff MJ; Starichkova VV; Trudgian TS, 2022, 'Explicit lower bounds on', Journal of Number Theory, 240, pp. 641 - 655, http://dx.doi.org/10.1016/j.jnt.2021.12.013
    Journal articles | 2022
    Mossinghoff MJ; Trudgian TS, 2022, 'Oscillations in the Goldbach conjecture', Journal de Theorie des Nombres de Bordeaux, 34, pp. 395 - 307, http://dx.doi.org/10.5802/jtnb.1202
    Journal articles | 2022
    Simonič A; TRUDGIAN TS; TURNAGE-BUTTERBAUGH CL, 2022, 'SOME EXPLICIT AND UNCONDITIONAL RESULTS ON GAPS BETWEEN ZEROES OF THE RIEMANN ZETA-FUNCTION', Transactions of the American Mathematical Society, 375, pp. 3239 - 3265, http://dx.doi.org/10.1090/tran/8571
    Journal articles | 2021
    Brent RP; Platt DJ; Trudgian TS, 2021, 'A HARMONIC SUM over NONTRIVIAL ZEROS of the RIEMANN ZETA-FUNCTION', Bulletin of the Australian Mathematical Society, 104, pp. 59 - 65, http://dx.doi.org/10.1017/S0004972720001252
    Journal articles | 2021
    Brent RP; Platt DJ; Trudgian TS, 2021, 'Accurate estimation of sums over zeros of the riemann zeta-function', Mathematics of Computation, 90, pp. 2923 - 2935, http://dx.doi.org/10.1090/mcom/3652
    Journal articles | 2021
    Cully-Hugill M; Trudgian T, 2021, 'Two explicit divisor sums', Ramanujan Journal, 56, pp. 141 - 149, http://dx.doi.org/10.1007/s11139-020-00333-3
    Journal articles | 2021
    MOSSINGHOFF MJ; SILVA TOE; TRUDGIAN TS, 2021, 'the Distribution of K-Free Numbers', Mathematics of Computation, 90, pp. 907 - 929, http://dx.doi.org/10.1090/MCOM/3581
    Journal articles | 2021
    Mossinghoff MJ; Trudgian TS, 2021, 'Oscillations in weighted arithmetic sums', International Journal of Number Theory, 17, pp. 1697 - 1716, http://dx.doi.org/10.1142/S1793042121500561
    Journal articles | 2021
    PLATT DJ; TRUDGIAN TS, 2021, 'the Error Term in the Prime Number Theorem', Mathematics of Computation, 90, pp. 871 - 881, http://dx.doi.org/10.1090/MCOM/3583
    Journal articles | 2021
    Platt D; Trudgian T, 2021, 'The Riemann hypothesis is true up to 3·1012', Bulletin of the London Mathematical Society, 53, pp. 792 - 797, http://dx.doi.org/10.1112/blms.12460
    Journal articles | 2020
    Kerr B; McGown KJ; Trudgian T, 2020, 'The least primitive root modulo p2', Journal of Number Theory, 215, pp. 20 - 27, http://dx.doi.org/10.1016/j.jnt.2020.01.002
    Journal articles | 2020
    Kobayashi M; Trudgian T, 2020, 'On integers n for which σ(2n + 1)≥σ(2n)', Journal of Number Theory, 215, pp. 138 - 148, http://dx.doi.org/10.1016/j.jnt.2020.02.007
    Journal articles | 2020
    McGown KJ; Trudgian T, 2020, 'Explicit upper bounds on the least primitive root', Proceedings of the American Mathematical Society, 148, pp. 1049 - 1061, http://dx.doi.org/10.1090/proc/14800
    Journal articles | 2019
    Bailey G; Cohen SD; Sutherland N; Trudgian T, 2019, 'Existence results for primitive elements in cubic and quartic extensions of a finite field', Mathematics of Computation, 88, pp. 931 - 947, http://dx.doi.org/10.1090/mcom/3357
    Journal articles | 2019
    Cohen SD; Trudgian T, 2019, 'Lehmer numbers and primitive roots modulo a prime', Journal of Number Theory, 203, pp. 68 - 79, http://dx.doi.org/10.1016/j.jnt.2019.03.004
    Journal articles | 2019
    Morrill T; Trudgian T, 2019, 'An elementary bound on Siegel zeroes', Journal of Number Theory, http://dx.doi.org/10.1016/j.jnt.2019.10.024
    Journal articles | 2019
    Platt D; Trudgian T, 2019, 'Fujii's development on Chebyshev's conjecture', International Journal of Number Theory, 15, pp. 639 - 644, http://dx.doi.org/10.1142/S1793042119500337
    Journal articles | 2018
    Booker AR; Cohen SD; Sutherland N; Trudgian T, 2018, 'Primitive values of quadratic polynomials in a finite field', Mathematics of Computation, 88, pp. 1903 - 1912, http://dx.doi.org/10.1090/mcom/3390
    Journal articles | 2018
    Cohen SD; Oliveira e Silva T; Sutherland N; Trudgian T, 2018, 'Linear combinations of primitive elements of a finite field', Finite Fields and their Applications, 51, pp. 388 - 406, http://dx.doi.org/10.1016/j.ffa.2018.02.009
    Journal articles | 2018
    Jarso T; Trudgian T, 2018, 'Quadratic non-residues that are not primitive roots', Mathematics of Computation, 88, pp. 1251 - 1260, http://dx.doi.org/10.1090/MCOM/3378
    Journal articles | 2018
    Munsch M; Trudgian T, 2018, 'Square-full primitive roots', International Journal of Number Theory, 14, pp. 1013 - 1021, http://dx.doi.org/10.1142/S1793042118500604
    Journal articles | 2017
    Cohen SD; Trudgian T, 2017, 'On the least square-free primitive root modulo p', Journal of Number Theory, 170, pp. 10 - 16, http://dx.doi.org/10.1016/j.jnt.2016.06.011
    Journal articles | 2017
    Platt DJ; Trudgian TS, 2017, 'On the sum of two squares and at most two powers of 2', American Mathematical Monthly, 124, pp. 737 - 740, http://dx.doi.org/10.4169/amer.math.monthly.124.8.737
    Journal articles | 2017
    Trudgian T; Wang Q, 2017, 'The T4 and G4 constructions of Costas arrays', Journal of Combinatorial Mathematics and Combinatorial Computing, 100, pp. 217 - 221
    Journal articles | 2016
    Cipu M; Trudgian T, 2016, 'Searching for Diophantine quintuples', Acta Arithmetica, 173, pp. 365 - 382, http://dx.doi.org/10.4064/aa8254-2-2016
    Journal articles | 2016
    Cohen SD; Oliveira E Silva T; Trudgian T, 2016, 'On Grosswald's conjecture on primitive roots', Acta Arithmetica, 172, pp. 263 - 270, http://dx.doi.org/10.4064/aa8109-12-2015
    Journal articles | 2016
    McGown K; Trevino E; Trudgian TS, 2016, 'Resolving Grosswald's conjecture on GRH', Functiones et Approximatio Commentarii Mathematici, 55, pp. 215 - 225
    Journal articles | 2016
    Platt DJ; Trudgian TS, 2016, 'Diophantine quintuples containing triples of the first kind', Periodica Mathematica Hungarica, 72, pp. 235 - 242, http://dx.doi.org/10.1007/s10998-016-0127-2
    Journal articles | 2016
    Platt DJ; Trudgian TS, 2016, 'On the first sign change of θ(x)-x', Mathematics of Computation, 85, pp. 1539 - 1547, http://dx.doi.org/10.1090/mcom/3021
    Journal articles | 2016
    Platt DJ; Trudgian TS, 2016, 'Zeroes of partial sums of the zeta-function', LMS Journal of Computation and Mathematics, 19, pp. 37 - 41, http://dx.doi.org/10.1112/S1461157015000340
    Journal articles | 2016
    Trudgian T, 2016, 'Updating the error term in the prime number theorem', Ramanujan Journal, 39, pp. 225 - 234, http://dx.doi.org/10.1007/s11139-014-9656-6
    Journal articles | 2016
    Trudgian TS, 2016, 'Improvements to Turing's Method II', Rocky Mountain Journal of Mathematics, 46, pp. 325 - 332
    Journal articles | 2015
    Akbary A; Trudgian TS, 2015, 'A log-free zero-density estimate and small gaps in coefficients of L-functions', International Mathematics Research Notices, 2015, pp. 4242 - 4268, http://dx.doi.org/10.1093/imrn/rnu065
    Journal articles | 2015
    Best DG; Trudgian TS, 2015, 'Linear relations of zeroes of the zeta-function', Mathematics of Computation, 84, pp. 2047 - 2058, http://dx.doi.org/10.1090/s0025-5718-2014-02916-5
    Journal articles | 2015
    Broughan K; Trudgian TS, 2015, 'Robin's inequality for 11-free integers', Integers: electronic journal of combinatorial number theory, 15
    Journal articles | 2015
    Cohen SD; Silva TO; Trudgian T, 2015, 'On consecutive primitive elements in a finite field', Bulletin of the London Mathematical Society, 47, pp. 418 - 426, http://dx.doi.org/10.1112/blms/bdv018
    Journal articles | 2015
    Cohen SD; Silva TOE; Trudgian T, 2015, 'A proof of the conjecture of Cohen and Mullen on sums of primitive roots', Mathematics of Computation, 84, pp. 2979 - 2986, http://dx.doi.org/10.1090/mcom/2950
    Journal articles | 2015
    Mossinghoff MJ; Trudgian TS, 2015, 'Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function', Journal of Number Theory, 157, pp. 329 - 349, http://dx.doi.org/10.1016/j.jnt.2015.05.010
    Journal articles | 2015
    Platt DJ; Trudgian TS, 2015, 'An improved explicit bound on', Journal of Number Theory, 147, pp. 842 - 851, http://dx.doi.org/10.1016/j.jnt.2014.08.019
    Journal articles | 2015
    Platt DJ; Trudgian TS, 2015, 'Linnik's approximation to Goldbach's conjecture, and other problems', Journal of Number Theory, 153, pp. 54 - 62, http://dx.doi.org/10.1016/j.jnt.2015.01.008
    Journal articles | 2015
    Saouter Y; Trudgian T; Demichel P, 2015, 'A still sharper region where π(x) - li(x) is positive', Mathematics of Computation, 84, pp. 2433 - 2446, http://dx.doi.org/10.1090/S0025-5718-2015-02930-5
    Journal articles | 2015
    Trudgian T, 2015, 'Bounds on the number of Diophantine quintuples', Journal of Number Theory, 157, pp. 233 - 249, http://dx.doi.org/10.1016/j.jnt.2015.05.004
    Journal articles | 2015
    Trudgian T, 2015, 'Explicit bounds on the logarithmic derivative and the reciprocal of the Riemann zeta-function', Functiones et Approximatio, Commentarii Mathematici, 52, pp. 253 - 261, http://dx.doi.org/10.7169/facm/2015.52.2.5
    Journal articles | 2015
    Trudgian T, 2015, 'The sum of the unitary divisor function', Publications de l'Institut Mathematique, 97, pp. 175 - 180, http://dx.doi.org/10.2298/PIM140617001T
    Journal articles | 2015
    Trudgian TS, 2015, 'A short extension of two of spira's results', Journal of Mathematical Inequalities, 9, pp. 795 - 798, http://dx.doi.org/10.7153/jmi-09-65
    Journal articles | 2015
    Trudgian TS, 2015, 'An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions', Mathematics of Computation, 84, pp. 1439 - 1450, http://dx.doi.org/10.1090/s0025-5718-2014-02898-6
    Journal articles | 2014
    Trudgian T, 2014, 'A new upper bound for ζ(1+ it)', Bulletin of the Australian Mathematical Society, 89, pp. 259 - 264, http://dx.doi.org/10.1017/S0004972713000415
    Journal articles | 2014
    Trudgian TS, 2014, 'An improved upper bound for the argument of the Riemann zeta-function on the critical line II', Journal of Number Theory, 134, pp. 280 - 292, http://dx.doi.org/10.1016/j.jnt.2013.07.017
    Journal articles | 2014
    Trudgian TS, 2014, 'There are no socialist primes less than 10^9', Integers: electronic journal of combinatorial number theory, 14
    Journal articles | 2012
    Mossinghoff MJ; Trudgian TS, 2012, 'Between the problems of Pólya and Turán', Journal of the Australian Mathematical Society, 93, pp. 157 - 171, http://dx.doi.org/10.1017/S1446788712000201
    Journal articles | 2012
    Trudgian T, 2012, 'An improved upper bound for the argument of the riemann zeta-function on the critical line', Mathematics of Computation, 81, pp. 1053 - 1061, http://dx.doi.org/10.1090/S0025-5718-2011-02537-8
    Journal articles | 2012
    Trudgian TS, 2012, 'On Selberg's method and multiplicities of the zeroes of the Riemann zeta-function', Commentarii Mathematici Universitatis Sancti Pauli, 60, pp. 227 - 229
    Journal articles | 2011
    Trudgian T, 2011, 'Improvements to Turing’s method', Mathematics of Computation, 80, pp. 2259 - 2259, http://dx.doi.org/10.1090/S0025-5718-2011-02470-1
    Journal articles | 2011
    Trudgian TS, 2011, 'On the success and failure of Gram's Law and the Rosser Rule', Acta Arithmetica, 148, pp. 225 - 256
    Journal articles | 2010
    Trudgian TS, 2010, 'On a conjecture of Shanks', Journal of Number Theory, 130, pp. 2635 - 2638, http://dx.doi.org/10.1016/j.jnt.2010.04.010
  • Preprints | 2022
    Bellotti C; Keller TM; Trudgian TS, 2022, New bounds for numbers of primes in element orders of finite groups, http://dx.doi.org/10.48550/arxiv.2211.05837
    Preprints | 2022
    Goldston DA; Trudgian TS; Turnage-Butterbaugh CL, 2022, On the Montgomery-Odlyzko method regarding gaps between zeros of the zeta-function, , http://dx.doi.org/10.48550/arxiv.2201.10676
    Preprints | 2022
    Mossinghoff MJ; Trudgian TS; Yang A, 2022, Explicit zero-free regions for the Riemann zeta-function, http://dx.doi.org/10.48550/arxiv.2212.06867
    Preprints | 2021
    Booker AR; Hathi S; Mossinghoff MJ; Trudgian TS, 2021, Wolstenholme and Vandiver primes, http://dx.doi.org/10.48550/arxiv.2101.11157
    Preprints | 2021
    Martin G; Mossinghoff MJ; Trudgian TS, 2021, Fake Mu's, http://dx.doi.org/10.48550/arxiv.2112.05227
    Preprints | 2021
    Mossinghoff MJ; Starichkova VV; Trudgian TS, 2021, Explicit lower bounds on $|L(1, χ)|$, , http://arxiv.org/abs/2107.09230v1
    Preprints | 2020
    Brent RP; Platt DJ; Trudgian TS, 2020, A harmonic sum over nontrivial zeros of the Riemann zeta-function, , http://dx.doi.org/10.48550/arxiv.2009.05251
    Preprints | 2020
    Brent RP; Platt DJ; Trudgian TS, 2020, Accurate estimation of sums over zeros of the Riemann zeta-function, , http://dx.doi.org/10.48550/arxiv.2009.13791
    Preprints | 2020
    Brent RP; Platt DJ; Trudgian TS, 2020, The mean square of the error term in the prime number theorem, , http://dx.doi.org/10.48550/arxiv.2008.06140
    Preprints | 2020
    Languasco A; Trudgian TS, 2020, Uniform effective estimates for $\vert L(1,\chi)\vert$, , http://dx.doi.org/10.48550/arxiv.2011.08348
    Preprints | 2020
    Mossinghoff MJ; Trudgian TS, 2020, Oscillations in weighted arithmetic sums, , http://dx.doi.org/10.48550/arxiv.2007.14537
    Preprints | 2020
    Mossinghoff MJ; Trudgian TS, 2020, The size of oscillations in the Goldbach conjecture, , http://dx.doi.org/10.48550/arxiv.2006.14742
    Conference Papers | 2020
    Platt D; Trudgian T, 2020, 'Improved bounds on Brun’s constant', in Springer Proceedings in Mathematics and Statistics, pp. 395 - 406, http://dx.doi.org/10.1007/978-3-030-36568-4_25
    Preprints | 2020
    Simonič A; Trudgian T; Turnage-Butterbaugh CL, 2020, Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function, , http://dx.doi.org/10.48550/arxiv.2010.10675
    Preprints | 2019
    Mossinghoff MJ; Trudgian TS, 2019, A tale of two omegas, , http://dx.doi.org/10.48550/arxiv.1906.02847
    Preprints | 2014
    Mossinghoff MJ; Trudgian TS, 2014, Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function, http://dx.doi.org/10.48550/arxiv.1410.3926
    Preprints | 2013
    Akbary A; Trudgian TS, 2013, A log-free zero-density estimate and small gaps in coefficients of $L$-functions, , http://dx.doi.org/10.48550/arxiv.1312.5820

DECRA, Australian Research Council, 2012-2015.

Discovery Project, Australian Research Council, 2015-2019 (joint with Dave Platt, Bristol).

Future Fellowship, Australian Research Council, 2016-2020.

Discovery Project, Australian Research Council, 2024-2027

My Research Supervision