# Dr Timothy Trudgian

## Areas of expertise

- Algebra And Number Theory 010101

## Biography

I graduated from the Australian National University in 2005, and received a General Sir John Monash Award to study overseas. In early 2010 I was awarded my DPhil from the University of Oxford, having studied under Prof. D.R. Heath-Brown.

I once took 6/24 in a college cricket match bowling an optimistic variety of medium-pace. I rather suspect that the opposition reversed their batting order.

During 2009-2010 I was a Lecturer in Mathematics at Merton College, Oxford. During 2010-2012 I was a postdoctoral research fellow at the University of Lethbridge.

## Researcher's projects

- Sign changes in some arithmetic functions (with Amir Akbary, Lethbridge)
- The first sign change in Mertens' formula (with Yannick Saouter, CNRS and Patrick Demichel, Hewlett-Packard France)
- Quadratic polynomials over primitive roots (with Stephen Cohen, Glasgow, and Tom\'{a}s Oliveira e Silva, Aveiro)
- Diophantine quintuples (with Dave Platt, Bristol)

## Available student projects

The following is a brief list of some honours and undergraduate projects. Please email me for more information.

Some general topics that provide an introduction to analytic number theory are

- The theory of the Riemann zeta-function
- Waring's problem
- Computational searches for specific primes

Some specific topics are

- Turan's power sum method
- The failure of the Mertens conjecture
- Explicit zero-free regions of the zeta-function and other L-functions
- Skewes' Number for Mertens' theorem

## Current student projects

- Adrian Dudek, PhD Student,
*Explicit estimates in number theory*, commenced November 2012 - Stijn Hanson, PhD Student,
*Almost-prime k-tuples,*commenced November 2014 - Jeffrey Lay, PhD Student,
*Oscillation Theorems*, commenced November 2014

## Past student projects

- Benedict Morrissey, Summer Research Scholar,
*Zero-density theorems for the Riemann zeta-function*, 2012-2013 - Shuhui He, Summer Research Scholar,
*Exploring exponent pairs,*2012-2013 - Eloise Hamilton, Summer Research Scholar,
*Explicit bounds on the nth prime number*, 2013-2014 - Nam Ho, Summer Research Scholar,
*Elliptic curves and triangle problems*, 2013-2014 - Jeffrey Lay, Honours Student,
*On the multiplicities of the zeroes of the Riemann zeta-function*, 2013-2014 - Kirsty Chalker, Summer Research Scholar,
*Zeroes of the Riemann zeta-function on the critical line*, 2014-2015 - Owen Hearder, Summer Research Scholar,
*Explicit divisor sums*, 2014-2015 - Kam-hung Yau, Summer Research Scholar,
*Gaps between zeroes of the Riemann zeta-function*, 2014-2015 - Madeleine Kyng, Summer Research Scholar,
*Consecutive primitive roots modulo a prime,*2015-2016 - Mitchell Chiew, Summer Research Scholar,
*Square-free numbers in short intervals*, 2015-2016 - Jakob Vtv, Summer Research Scholar,
*Consecutive square-free numbers*, 2015-2016

## Publications

- Trudgian, T 2015, 'Updating the error term in the prime number theorem', Ramanujan Journal, vol. Online Early Version, no. 2015, pp. 1-10.
- Trudgian, T 2015, 'Improvements to Turing's method II', Rocky Mountain Journal of Mathematics, vol. Online Early Version, no. 2015, pp. 1-7.
- Trudgian, T 2015, 'An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions', Mathematics of Computation, vol. 84, no. 293, pp. 1439-1450.
- Broughan, K & Trudgian, T 2015, 'Robin's Inequality for 11-free Integers', Integers, vol. 15, no. A12, 5pp.
- Platt, D & Trudgian, T 2015, 'An improved explicit bound on $|\zeta(1/2 + it)|$', Journal of Number Theory, vol. 147, pp. 842-851
- Trudgian, T 2015, 'A short extension of two of Spira's results', Journal of Mathematical Inequalities, vol. 9, no.3, pp. 795-798.
- Trudgian, T 2015, 'Explicit bounds on the logarithmic derivative and the reciprocal of the Riemann zeta-function', Functiones et Approximatio, Commentarii Mathematici, vol. 52, no. 2, pp. 253-261
- Trudgian, T 2015, 'The sum of the unitary divisor function', Publications de l'Institut Mathematique (Beograd) (N.S.), vol. 97, no. 111, pp. 175-180.
- Platt, D & Trudgian, T 2014, 'Linnik's approximation to Goldbach's conjecture, and other problems', Journal of Number Theory, vol. 153, pp. 54-62.
- Trudgian, T 2014, 'There are no socialist primes less than 10^9', Integers, vol. 14, no. A63, pp. 1-4.
- Trudgian, T 2014, 'An Improved Upper Bound for the Argument of the Riemann zeta-function on the critical line II', Journal of Number Theory, vol. 134, pp. 280-292.
- Best, D & Trudgian, T 2014, 'Linear relations of zeroes of the zeta-function', Mathematics of Computation, vol. 84, no. 294, pp. 2047-2058.
- Akbary, A & Trudgian, T 2015, 'A Log-Free Zero-Density Estimate and Small Gaps in Coefficients of L-Functions', International Mathematics Research Notices, no. 12, pp.\ 4242-4268
- Cohen, S, Oliveira e Silva, T and Trudgian, T 2015, A proof of the conjecture of Cohen and Mullen on sums of primitive roots, Mathematics of Computation, vol. 84, no. 296, pp. 2979-2986
- Saouter, Y, Demichel, P and Trudgian, T 2015, A still sharper region where $\pi(x) - \textrm{li}(x)$ is positive, Math. Comp., vol. 84, no. 295, pp. 2433-2446
- Trudgian, T 2014, 'A new upper bound for $|\zeta(1+ it)|$', Bulletin of the Australian Mathematical Society, vol. 89, no. 2, pp. 259-264.
- Trudgian, T 2013, 'Twin progress in number theory', Australian Mathematical Society Gazette, vol. 40, no. 3, pp. 202-207.
- Trudgian, T 2012, 'An improved upper bound for the argument of the Riemann zeta-function on the critical line', Mathematics of Computation, vol. 81, no. 278, pp. 1053-1061.
- Mossinghoff, M & Trudgian, T 2012, 'Between the problems of Polya and Turan', J. Austral. Math. Soc., vol. 93, iss. 1-2, pp. 157-171
- Trudgian, T 2011, 'Selberg's method and the multiplicities of the zeroes of the Riemann zeta-function', Commentarii Mathematici Universitatis Sancti Pauli, vol. 60, no. 1-2, pp. 227-229.
- Trudgian, T 2011, 'Improvements to Turing's Method', Math. Comp., vol. 80, pp. 2259-2279
- Trudgian, T 2011, 'On the success and failure of Gram's Law and the Rosser Rule', Acta Arith., vol. 148, no. 3, pp. 225-256
- Trudgian, T 2010, 'On a Conjecture of Shanks', J. Number Theory, vol.130, iss.12, pp. 2635-2638
- Trudgian, T 2009, 'Introducing Complex Numbers', The Australian Senior Mathematics Journal, vol. 23, no. 2, pp. 59-62

## Projects and Grants

Grants are drawn from ARIES. To add Projects or Grants please contact your College Research Office.

- Verifying the Riemann hypothesis to large heights: theory and applications (Primary Investigator)
- A new upper bound for the Riemann zeta-function and applications to the distributions of prime numbers (Primary Investigator)