156 citations · 177 across the 20 of their papers we have counts for
26 papers
Functional Equations Characterize Dirichlet Characters
Ghaith Hiary, Ali Saraeb
We prove a converse theorem for functional equations of Dirichlet -functions. Under mild assumptions, we prove that these functional equations for -series of the form $\sum_{…
Unconditional estimates on the argument of Dirichlet -functions with applications to low-lying zeros
Ghaith Hiary, Tianyu Zhao
We make explicit a result of Selberg on the argument of Dirichlet -functions averaged over non-principal characters modulo a prime . As a corollary, we show for all sufficien…
A method for verifying the generalized Riemann hypothesis
Ghaith Hiary, Summer Ireland, Megan Kyi
Riemann numerically approximated at least three zeta zeros. According to Edwards, Riemann even took steps to verify that the lowest zero he computed was indeed the first zeta zero.…
Counting sign changes of partial sums of random multiplicative functions
Nick Geis, Ghaith Hiary
Let be a Rademacher random multiplicative function. Let be the partial sum of . Let denote the number of sign changes of u…
Explicit bounds for the Riemann zeta-function on the 1-line
Ghaith A. Hiary, Nicol Leong, Andrew Yang
Explicit estimates for the Riemann zeta-function on the -line are derived using various methods, in particular van der Corput lemmas of high order and a theorem of Borel and Car…
A Generalization of Lehman's Method
Jonathon Hales, Ghaith Hiary
A new deterministic algorithm for finding square divisors, and finding -power divisors in general, is presented. This algorithm is based on Lehman's method for integer factoriza…
The Legendre Approximation and Arithmetic Bias
Ghaith Hiary, Megan Paasche
An interesting episode in the history of the prime number theorem concerns a formula proposed by Legendre for counting the primes below a given bound. We point out that arithmetic…
New Formulas for the Riemann Zeta Function
Aditya Akula, Ghaith Hiary
A new method for continuing the usual Dirichlet series that defines the Riemann zeta function is presented. Numerical experiments demonstrating the computational efficacy of…
An improved explicit estimate for
Ghaith A. Hiary, Dhir Patel, Andrew Yang
An explicit subconvex bound for the Riemann zeta function on the critical line is proved. Previous subconvex bounds relied on an incorrect version of the Kusmin-L…
Calculations of the invariant measure for Hurwitz Continued Fractions
Ghaith Hiary, Joseph Vandehey
We study the density of the invariant measure of the Hurwitz complex continued fraction from a computational perspective. It is known that this density is piece-wise real-analytic…
Asymptotics and formulas for cubic exponential sums
Ghaith A. Hiary
Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generali…
A fast amortized algorithm for computing quadratic Dirichlet -functions
Ghaith A. Hiary
An algorithm to compute Dirichlet -functions for many quadratic characters is derived. The algorithm is optimal (up to logarithmic factors) provided that the conductors of the c…
New computations of the Riemann zeta function on the critical line
Jonathan W. Bober, Ghaith A. Hiary
We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the sec…
An explicit hybrid estimate for
Ghaith A. Hiary
An explicit hybrid estimate for is derived, where is a Dirichlet character modulo . The estimate applies when is bounded away from zero, and is most effect…
An explicit van der Corput estimate for
Ghaith A. Hiary
An explicit estimate for the Riemann zeta function on the critical line is derived using the van der Corput method. An explicit van der Corput lemma is presented.
A deterministic algorithm for integer factorization
Ghaith A. Hiary
A deterministic algorithm for factoring using bit operations is presented. The algorithm tests the divisibility of by all the integers in a short interval at…
An alternative to Riemann-Siegel type formulas
Ghaith A. Hiary
Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet -functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geo…
Detecting squarefree numbers
Andrew R. Booker, Ghaith A. Hiary, Jon P. Keating
We present an algorithm, based on the explicit formula for -functions and conditional on GRH, for proving that a given integer is squarefree with little or no knowledge of its f…
Computing Dirichlet character sums to a power-full modulus
Ghaith A. Hiary
The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an…
Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function
Yan V. Fyodorov, Ghaith A Hiary, Jonathan P. Keating
We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the ma…
Uniform asymptotics for the full moment conjecture of the Riemann zeta function
Ghaith A. Hiary, Michael O. Rubinstein
Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of -functions. In the case of the Riemann zeta function, t…
Numerical study of the derivative of the Riemann zeta function at zeros
Ghaith A. Hiary, Andrew M. Odlyzko
The derivative of the Riemann zeta function was computed numerically on several large sets of zeros at large heights. Comparisons to known and conjectured asymptotics are presented…
The zeta function on the critical line: Numerical evidence for moments and random matrix theory models
Ghaith A. Hiary, Andrew M. Odlyzko
Results of extensive computations of moments of the Riemann zeta function on the critical line are presented. Calculated values are compared with predictions motivated by random ma…
An amortized-complexity method to compute the Riemann zeta function
G. A. Hiary
A practical method to compute the Riemann zeta function is presented. The method can compute at any points in using an average…