Ghaith A. Hiary
I am an assistant professor in the Department of Mathematics at the Ohio State University.
I graduated from the
University of Minnesota, Minneapolis,
with a PhD in Mathematics in August 2008, supervised by
Andrew Odlyzko. My research is supported by a grant from the National Science Foundation DMS1406190.
You can reach me
at hiary.1@osu.edu or hiaryg@gmail.com.
Teaching 20172018
Publications and preprints
Note. The arXiv version 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), 17231752.

A nearlyoptimal method to compute the truncated theta function, its derivatives, and integrals,
Ann. Math., 1742 (2011) 859889.

Fast methods to compute the Riemann zeta function,
Ann. Math., 1742 (2011) 891946.

An amortizedcomplexity method to compute the Riemann zeta function,
Math. Comp., 80 (2011) 17851796.

Numerical study of the derivative of the Riemann zeta function at zeros,
with A. M. Odlyzko, Commentarii Mathematici Universitatis Sancti Pauli, vol. 60, no. 12, (2011) 4760.

Uniform asymptotics of the coefficients of unitary moment polynomials,
with M. O. Rubinstein, Proc. R. Soc. A., vol. 467, no. 2128, (2011) 10731100.

Uniform asymptotics for the full moment conjecture of the Riemann zeta function,
with M. O. Rubinstein, J. Number Theor., Volume 132, Issue 4, (2012), 820868.

Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann ZetaFunction, with Y. V. Fyodorov and and J. P. Keating, Phys. Rev. Lett. 108, 170601 (2012).

Computing Dirichlet character sums to a powerfull modulus, J. Number Theor. 140 (2014), pp. 122146.

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), 211239.

New computations of the Riemann zeta function on the critical line with Jonathan Bober, accepted in Experimental Mathematics.

A fast amortized algorithm for computing quadratic Dirichlet Lfunctions, accepted in Quarterly Journal of Mathematics.

Asymptotics and formulas for cubic exponential sums.
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 RiemannSiegel 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 scripts to orgnize the multiprocess computation. It also includes a separate
program to extract "zeta data" e.g. derivative, max,... using bandlimited 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

I'm developing a library for fast computations with Dirichlet
and other
Lfunctions. It consists of various
algorithms that I have developed
over the years, with a particular focus on largescale computations.
See here for an example.
This is a fairly longterm project.
Past teaching

Math 5591H & 5112: Honors Abstract Algebra (OSU, Spring 2017)

Math 5590H & 5111: Honors Abstract Algebra (OSU, Fall 2016)

Math 8120: Computational Number Theory (OSU, Spring 2016)

Math 2568: Linear Algebra (OSU, Spring 2016)

Math 5152: Introduction to Number Theory with Applications (OSU, Spring 2016)

Math 6193: Individual Studies Numerical Linear Algebra (OSU, Spring 2016)

Math 4193: Individual Studies (OSU, Fall 2015).

Math 2177: Mathematical Topics for Engineers (OSU, Spring 2015).

Math 5603: Numerical linear Algebra (OSU, Fall 2014).

Math 2173: Engineering Mathematics B (OSU, Spring 2014).

Math 5601: Essentials of numerical methods (OSU, Fall 2013).

Computational Mathematics, Mechanics, and Calculus tutorials (Bristol, Fall 2012  Spring 2013).

PM 340: Elementary number theory (Waterloo, Fall 2010).

Calculus I, Calculus II, Multivariable Calculus, Linear Algebra and Differential Equations recitations (UMN, Fall 2002  Spring 2006)
Seminar
OSU Number Theory Seminar
Some slides
Misc
Pictures