arXiv subjects
Publications and source records attributed to Ghaith Hiary.
We prove a converse theorem for functional equations of Dirichlet $L$-functions. Under mild assumptions, we prove that these functional equations for $L$-series of the form $\sum_{n\ge 1} f(n) n^{-s}$ force the coefficient function $f$ to be a primitive Dirichlet character. Consequently, these functional equations force the existence of an Euler product.
We make explicit a result of Selberg on the argument of Dirichlet $L$-functions averaged over non-principal characters modulo a prime $q$. As a corollary, we show for all sufficiently large prime $q$ that the height of the lowest non-trivial zero of the corresponding family of $L$-functions is less than $1075\cdot \frac{2\pi}{\log q}$. Here the scaling factor $\frac{2\pi}{\log q}$ is the average spacing between consecutive low-lying zeros with height at most 1, say. We also obtain a lower bound on the proportion of $L$-functions whose first zero lies within a given multiple of the average spacing. These appear to be the first explicit unconditional results of their kinds.
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. This approach to verification is developed, improved, and generalized to a large class of $L$-functions. Results of numerical calculations demonstrating the efficacy of the method are presented.
Let $f$ be a Rademacher random multiplicative function. Let $$M_f(u):=\sum_{n \leq u} f(n)$$ be the partial sum of $f$. Let $V_f(x)$ denote the number of sign changes of $M_f(u)$ up to $x$. We show that for any constant $c > 2$, $$V_f(x) = \Omega ((\log \log \log x)^{1/c} )$$ almost surely.
A new deterministic algorithm for finding square divisors, and finding $r$-power divisors in general, is presented. This algorithm is based on Lehman's method for integer factorization and is straightforward to implement. While the theoretical complexity of the new algorithm is far from best known, the algorithm becomes especially effective if even a loose bound on a square divisor is known. Additionally, we answer a question by D. Harvey and M. Hittmeir on whether their recent deterministic algorithm for integer factorization can be adapted to finding $r$-power divisors.
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 bias likely played an important role in arriving at that formula and in its subsequent widespread, decades-long recognition. We also show that the Legendre constant 1.08366 satisfies a certain simple and natural criterion, and conjecture that this criterion is how Legendre arrived at that erroneous constant in his formula.
A new method for continuing the usual Dirichlet series that defines the Riemann zeta function $ζ(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.
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 and so we provide a method for calculating the Taylor coefficients around certain points and also the results of our calculations. While our method does not find a simple "closed form" for the density of the invariant measure (if one even exists), our work leads us to some new conjectures about the behavior of the density at certain points. In addition to this, we detail all admissible strings of digits in the Hurwitz expansion. This may be of independent interest.