Address: Dept. of Mathematics
231 West 18th Avenue
Columbus, Ohio 43210-1174

Office:	MW 646
Phone :	(614) 292-4013
Email :	hiary.1@osu.edu

I am a professor in the Department of Mathematics at the Ohio State University. You can reach me at hiary.1@osu.edu

• Other name spelling: Gaith A. Hiary
• Pronunciation:

Publications and preprints

Note. The arXiv version most likely differs from published version.
• The zeta function on the critical line: numerical evidence for moments and random matrix theory models, with A. M. Odlyzko, Math. Comp. 81 (2012), no. 279, 1723–1752.
• A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals, Ann. of Math. (2) 174 (2011), no. 2, 859–889.
• Fast methods to compute the Riemann zeta function, Ann. of Math. (2) 174 (2011), no. 2, 891–946.
• An amortized-complexity method to compute the Riemann zeta function, Math. Comp. 80 (2011), no. 275, 1785–1796.
• Numerical study of the derivative of the Riemann zeta function at zeros, with A. M. Odlyzko, Comment. Math. Univ. St. Pauli 60 (2011), no. 1-2, 47–60.
• Uniform asymptotics of the coefficients of unitary moment polynomials, with M. O. Rubinstein, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2128, 1073–1100.
• Uniform asymptotics for the full moment conjecture of the Riemann zeta function, with M. O. Rubinstein, J. Number Theory 132 (2012), no. 4, 820–868.
• Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function, with Y. V. Fyodorov and J. P. Keating, Phys. Rev. Lett. 108, 170601 (2012).
• Computing Dirichlet character sums to a power-full modulus, J. Number Theory 140 (2014), 122–146.
• Detecting squarefree numbers with A. R. Booker and J. P. Keating, Duke Math. J. 164 (2015), no. 2, 235–275.
• An alternative to Riemann-–Siegel type formulas, Math. Comp. 85 (2016), no. 298, 1017–1032.
• A deterministic algorithm for integer factorization, Math. Comp. 85 (2016), no. 300, 2065–2069.
• An explicit van der Corput estimate for zeta(1/2+it), Indag. Math. (N.S.) 27 (2016), no. 2, 524–533.
• An explicit hybrid estimate for L(1/2+it, chi), Acta Arithmetica 176 (2016), 211-239.
• Asymptotics and formulas for cubic exponential sums, Proc. Steklov Inst. Math. (2017) 299, pp 78-95.
• A fast amortized algorithm for computing quadratic Dirichlet L-functions, Q. J. Math. 69 (2018), no.1, 1–12.
• New computations of the Riemann zeta function on the critical line with Jonathan Bober, Exp. Math. 27 (2018), no.2, 125–137.
• Calculations of the invariant measure for Hurwitz Continued Fractions with Joseph Vandehey, Exp. Math. 31 (2022), no.1, 324–336.
• New Formulas for the Riemann Zeta Function, with A. Akula, Integers 23 (2023), Paper No. A21, 16 pp.
• An improved explicit estimate for $\zeta(1/2+it)$, with D. Patel and A. Yang. Journal of Number Theory, Volume 256 (2024), Pages 195-217.
• The Legendre Approximation and Arithmetic Bias, with M. Paasche. Minnesota Journal of Undergraduate Mathematics, Vol. 8 No. 1 (2024).
• A Generalization of Lehman's Method, with J. Hales. Ramanujan Journal, Volume 65, pages 1773–1790, (2024).
• Explicit bounds for the Riemann zeta-function on the 1-line, with N. Leong and A. Yang. Funct. Approx. Comment. Math. Advance Publication 1-37 (2025).
• Counting sign changes of partial sums of random multiplicative functions, with N. Geis. The Quarterly Journal of Mathematics, Volume 76, Issue 1, March 2025, Pages 19–45.
• A method for verifying the generalized Riemann hypothesis, with S. Ireland and M. Kyi, Accepted in Math. Comp.
• Unconditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros, with T. Zhao.

Code, data, and experiments

• An explicit van der Corput bound for $\zeta(1/2+it)$ (paper here).

  Mathematica notebook to verify the van der Corput bound on the Riemann zeta function

• An alternative to Riemann--Siegel type formulas (paper here).

  Mathematica notebook containing a basic implementation

• A Deterministic $n^{1/3+o(1)}$ integer factoring algorithm (paper here).

  Mathematica notebook containing a basic implementation

• I've implemented the amortized complexity algorithm to compute zeta, described here. The implementation is in C++, aided by python/Sage scripts to orgnize the multi-process computation. It also includes a separate Sage program to extract "zeta data" -e.g. derivative, max,...- using band-limited interpolation. A database of (by now) ~600 million zeros near t = 10^28 (as well as smaller sets of about 200 million zeros at lower heights) has been obtained using the amortized algorithm, together with "raw data" files that allow quick extraction of further data that might be of interest. The computation took a few months on the riemann machine at U. Waterloo.

  Sample data for the zeta function obtained using the amortized algorithm

• This is a previous coding collaboration, with Jonathan Bober to implement my $T^{1/3+o(1)}$-algorithm to compute zeta, which is described here, here, and here. The implementation is in C++. It was quite useful during the implementation to constantly compare answers obtained from the C++ code with answers obtained from a basic version of the algorithm that I implemented in Mathematica back in 2009. The implementation essentially consisted of coding up the various formulas that were specified in the papers describing the algorithm.

  Description of the algorithm and some numerical results

Seminar

OSU Number Theory Seminar

Some slides

• Arithmetic biases
• Detecting square-free numbers via the explicit formula
• Large values and random matrices
• Fast methods to compute the Riemann zeta function
Pictures