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 an associate professor in the Department of Mathematics at the Ohio State University. You can reach me at hiary.1@osu.edu or hiaryg@gmail.com.

Teaching

• Math 6111: Abstract algebra I (OSU, Autumn 2019)

Publications and preprints

• The zeta function on the critical line: numerical evidence for moments and random matrix theory models, with A. M. Odlyzko, Math. Comp. 81 (2012), 1723-1752.
• A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals, Ann. Math., 174-2 (2011) 859-889.
• Fast methods to compute the Riemann zeta function, Ann. Math., 174-2 (2011) 891-946.
• An amortized-complexity method to compute the Riemann zeta function, Math. Comp., 80 (2011) 1785-1796.
• Numerical study of the derivative of the Riemann zeta function at zeros, with A. M. Odlyzko, Commentarii Mathematici Universitatis Sancti Pauli, vol. 60, no. 1-2, (2011) 47-60.
• Uniform asymptotics of the coefficients of unitary moment polynomials, with M. O. Rubinstein, Proc. R. Soc. A., vol. 467, no. 2128, (2011) 1073-1100.
• Uniform asymptotics for the full moment conjecture of the Riemann zeta function, with M. O. Rubinstein, J. Number Theor., Volume 132, Issue 4, (2012), 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 Theor. 140 (2014), pp. 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, The Quarterly Journal of Mathematics, Volume 69, Issue 1, Pages 1–12.
• New computations of the Riemann zeta function on the critical line with Jonathan Bober, Experimental Mathematics, 27:2 (2018), 125-137.
• Calculations of the invariant measure for Hurwitz Continued Fractions with Joseph Vandehey, Experimental Mathematics, (2019).

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

• I'm developing a library for fast computations with Dirichlet and other L-functions. It consists of various algorithms that I have developed over the years, with a particular focus on large-scale computations. See here for an example. This is a fairly long-term project.
• Math software packages of possible interest:

  SageMath (free, open-source, well-documented, comprehensive, suited for teaching and research)
  CoCalc (free, open-source, well-documented, web-based cloud computing, targeted for teaching)
  LMFDB (pioneering online database for L-functions)
  Wolfram Alpha (web-based, especially easy to use)
  Mathematica (well-documented, comperehensive, suited for teaching and research)
  Matlab (well-documented, easy to learn, suited for teaching and research especially in numerical linear algebra)
  R (free, open-source, well-documented, my favorite statistical package!)
  Pari (free software for number theory compuations and more.)
  lcalc (free software for L-function computations.)

  Also, Vim is a powerful text editor, and awk is a useful scripting language in command-line.

Past teaching

• Math 3345: Foundations of Higher Mathematics (OSU, Spring 2019)
• Math 7122: Analytic Number Theory (OSU, Spring 2019)
• Math 3345: Foundations of Higher Mathematics (OSU, Autumn 2018)
• Math 2153: Calculus III (OSU, Spring 2018)
• 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

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