Research

My area of research is analytic number theory.  I am interested in the Riemann zeta function, L-functions, prime numbers, and a variety of problems in multiplicative number theory. My Ph.D. thesis, Limiting Distributions and Zeros of Artin L-functions, studied the finer behaviour of the summatory function of the Möbius function and Chebyshev's bias in Galois groups. Some research topics I work on include: mean values of L-functions, non-vanishing of L-functions, sums over the zeros of the zeta function, gaps between zeros of the zeta function, simple zeros of L-functions, linear combinations of zeros of L-functions, explicit number theory, and various questions concerning distribution functions of number theoretic functions. In the past ten years, I have extensively studied moments of the Riemann zeta function, mean values of long Dirichlet polynomials, and convolution sums of arithmetic functions (see articles 7,8,9,10,12,15). I have also been actively involved in establishing a number of results in explicit number theory including explicit zero-density results and explicit bounds for prime counting functions (see articles 1,11,18).

1. An effective version of Chebotarev's density theorem
   with Sourabhashis Das and Habiba Kadiri.
   arXiv preprint: arXiv:2508.09480  

2. Prime Number Error Terms
   arXiv preprint: arXiv:2505.11295

3. Conference for Humans
   with A Hamieh, H Kadiri, G Martin,
   CMS Notes 57 (2025), no. 3, 33–46. 

4. Moments of L-functions Problem List
   with A Hamieh, H Kadiri, G Martin
   arXiv preprint arXiv:2405.17800

5. Comparative Prime Number Theory Problem List
   with A Hamieh, H Kadiri, G Martin
   arXiv preprint arXiv:2407.03530

6. A discrete mean value of the Riemann zeta function
   with K Benli, E Elma,
   Mathematika 70 (2024), no. 4, Paper No. e12281, 47 pp.
   arXiv preprint arXiv:2311.13554

7. Mean values of long Dirichlet polynomials with divisor coefficients
   with F Cicek, A Hamieh, to appear,
   Algebra and Number Theory.
   arXiv preprint arXiv:2309.08057

8. Mean values of long Dirichlet polynomials with higher divisor coefficients
   with A Hamieh
   Advances in Mathematics 410 (2022), part B, Paper No. 108759, 61 pp.
   arXiv preprint arXiv:2105.03525

9. The eighth moment of the Riemann zeta function
   with Q Shen, PJ Wong, to appear,
   Journal of the European  Mathematical Society.
   arXiv preprint arXiv:2204.13891

10. Shifted moments of the Riemann zeta function
    with Q Shen, PJ Wong,
    Canadian Journal of Mathematics 76 (2024), no. 5, 1556–1586.
    arXiv preprint arXiv:2206.03350

11. Sharper bounds for the Chebyshev function 𝜃 (𝑥)
    S Broadbent, H Kadiri, A Lumley, N Ng, K Wilk,
    Mathematics of Computation 90 (2021), no. 331, 2281–2315.
    arXiv preprint: arXiv:2002.11068

12. The sixth moment of the Riemann zeta function and ternary additive divisor sums,
    Discrete Analysis 2021, Paper No. 6, 60 pp.
    arXiv and final version: arXiv:1610.04977

13. Inclusive prime number races
    with G Martin,
    Transactions of the American Mathematical Society 373 (2020), no. 5, 3561–3607.
    arXiv preprint: arXiv:1710.00088

14. The least prime ideal in the Chebotarev density theorem
    with H Kadiri, PJ Wong,
    Proceedings of the American Mathematical Society 147 (2019), no. 6, 2289–2303.
    arXiv preprint: arXiv:1902.08640

15. Bounds and conjectures for additive divisor sums
    with M Thom,
    Functiones et Approximatio Commentarii Mathematici  60 (2019), no. 1, 97–142.
    arXiv preprint: arXiv:1609.01411

16. Subconvexity for modular form L-functions in the t aspect
    with AR Booker and MB Milinovich,
    Advances in Mathematics 341 (2019), 299–335.
    arXiv preprint: arXiv:1707.01576

17. Quantitative estimates for simple zeros of  L-functions
    with AR Booker and MB Milinovich,
    Mathematika 65 (2), 375-399, 2019.
    arXiv preprint: arXiv:1806.01959

18. Explicit zero density for the Riemann zeta function
    with H Kadiri and A Lumley,
    Journal of Mathematical Analysis and Applications 465 (1), 22-46, 2018.
    arXiv preprint: arXiv:2101.12263

19. Limiting distributions of the classical error terms of prime number theory
    with A Akbary and M Shahabi,
    The Quarterly Journal of Mathematics 65 (3), 743-780, 2014.
    arXiv preprint: arXiv:1306.1657

20. Lower bounds for moments of zeta prime rho
    with MB Milinovich,
    International Mathematics Research Notices 12, 3190-3216, 2014.

21. Nonzero values of Dirichlet L-functions in vertical arithmetic progressions
    with G Martin,
    International Journal of Number Theory 9 (04), 813-843, 2013.
    arXiv preprint: arXiv:1109.1788 

22. Simple zeros of modular L-functions
    with MB Milinovich,
    Proceedings of the London Mathematical Society 109 (6), 1465-1506, 2013.
    arXiv preprint: arXiv:1306.0854

23. A note on a conjecture of Gonek,
    with MB Milinovich,
    Functiones et Approximatio Commentarii Mathematici 46 (2), 177-187, 2012.
    arXiv preprint: arXiv:1106.1160

24. Explicit zero density theorems for Dedekind zeta functions
    with H Kadiri,
    Journal of Number Theory 132 (4), 748-775, 2012.
    arXiv preprint: arXiv:1201.3932

25. A note on the gaps between consecutive zeros of the Riemann zeta-function
    with HM Bui and MB Milinovich,
    Proceedings of the American Mathematical Society 138 (12), 4167-4175, 2010.
    arXiv preprint: arXiv:0910.2052

26. The diophantine equation A^4+ 2δB^2= C^n
    with MA Bennett and JS Ellenberg,
    International Journal of Number Theory  6 (2), 311-338, 2010.

27. Extreme values of ζ′(ρ)
    Journal of the London Mathematical Society 78 (2), 273-289, 2008.
    arXiv preprint: arXiv:0706.1765

28. Large gaps between the zeros of the Riemann zeta function
    Journal of Number Theory 128 (3), 509-556, 2008.
    arXiv preprint: arXiv:math/0510530

29. The Möbius function in short intervals,
    CRM Proc. Lect. Notes 46, 247-258, 2008.

30. A discrete mean value of the derivative of the Riemann zeta function,
    Mathematika 54 (1-2), 113-155, 2007.
    arXiv preprint: arXiv:0706.1763

31. The fourth moment of ζ′(ρ)
    Duke Mathematical Journal 125 (2), 243-266, 2004.
    arXiv preprint: arXiv:math/0310382

32. The distribution of the summatory function of the Möbius function
    Proceedings of the London Mathematical Society 89 (2), 361-389, 2004.
    arXiv preprint: arXiv:math/0310381

33. The distribution of values of the summatory function of the Möbius function,
    Canadian Mathematical Society Notes, 2002.

34. Limiting distributions and zeros of Artin L-functions
    PhD thesis, University of British Columbia, 2000.

35.  Two appendices in the article:
    Ram Murty, M.,Some remarks on the Riemann hypothesis,
    Tata Inst. Fund. Res. Stud. Math., 15, Cohomology of arithmetic groups, L-functions and automorphic forms
    (Mumbai, 1998/1999), 180–196, Tata Inst. Fund. Res., Bombay, 2001.

1. An effective version of Chebotarev's density theorem II
   with Sourabhashis Das and Habiba Kadiri.

2. Bounds for Mertens sums
   with  Samuel Broadbent, Andrew Fiori, Habiba Kadiri, and Kirsten Wilk.

3. An Explicit Version of Linnik's theorem for the least prime in an arithmetic progression
   with Cruz Castillo, Michaela Cully-Hugill, Jewel Mahajan, and  Chi Hoi Yip.

4. The Fourth discrete moment of the Riemann zeta function
   with Kubra Benli, Fatma Cicek, Ertan Elma, and Alia Hamieh. 

5. Scaling the ordinates of the zeta function
   with William Banks, Greg Martin, and Micah Milinovich.