Sep 2026· Proceedings of the American Mathematical Society· Vol 153, pp. 611-623· 0 citations· 26 references
Mathematics
Abstract
Motivated by the questions posed by W. V. Li and A. Wei and the conjecture of E. Lundberg and A. Thomack, we study the expected number of zeros of random harmonic polynomials $H_{n,m}(z)=p_n(z)+\overline{q_m(z)}$ with independently and identically distributed Gaussian coefficients. In this paper we verify the conjecture of E. Lundberg and A. Thomack that the expectation is $O(n)$ when $\deg p=\alpha\deg q$, where $0\leq\alpha<1$. This result extends the previous estimates when $m$ is a fixed constant or $m=n$ to more general case.
In this paper, we study a Riesz-type weighted sum of the Liouville function, $f(x):=-\sum_{n\le x}\frac{\lambda(n)\log n}{\sqrt n}\log\frac{x}{n}$ for sufficiently large $x$. Motivated by a recent research on the sign criteria for the Riemann Hypothesis arising from weighted prime-counting functions, we investigate the...
We study the family of functions $V_z:s \mapsto \zeta(s) \exp (zs)$ with numerical experiments. For selected sequences $z_n=x_nu$ on rays, our experiments suggest that suitable fixed points of $V_{z_n}$ approach nontrivial zeros of $\zeta$ in a nearly logarithmic pattern. We formulate this behavior as conjectures with...
Let $p_n$ be a random polynomial of degree $n$ whose roots are independent and identically distributed according to a rotationally invariant probability measure $\mu_0$ on the complex plane with finite logarithmic moment. If $k_n/n\to t\in(0,1)$, we prove that, as $n \to \infty$, the empirical zero measure of the $k_n$...
We study the distribution of the zeros of $\det P_N(z)$ where $P_N(z)$ is a random monic polynomial matrix, i.e., $P_N(z)=z^dI-\sum_{j=0}^{d-1}A_{j,N}z^j$ for possibly coupled random matrices $A_{j,N}$, scaled to have entrywise variance $O(1/N)$. We provide general conditions under which this distribution almost-surely...
For elliptic curves $E/\mathbb{Q}$ in short Weierstrass form \[ E=E(a_4,a_6): y^2=x^3+a_4x+a_6, \] we derive a multivariable Euler product generating function which encodes the Tamagawa product $Tam(E)=\prod_p c_p(E)$. Using this generating function, we compute limiting distributions, exact covariances, and moment and...
David Kurniadi Angdinata, Kenny Lau, Ken Ono et al.· 0 citations
We prove a one-sided central limit theorem for $\log |L(\tfrac 12,\chi)|$ in a thin family of $n$-th order Hecke characters, for $n \geq 3$. The proof combines the method of Radziwi{\l}{\l} and Soundararajan with a recent asymptotic formula of Diaconu, Ion, Pa\c{s}ol, and Popa for the twisted second moment with a power...
Lewis McGoldrick· 0 citations
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