We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree n, with not necessarily independent coefficients, decays like √ logn/n. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of t...

The expected number of real zeros of the polynomial of the form a0 a1x a2x · · · anx, where a0, a1, a2, . . . , an is a sequence of standard Gaussian random variables, is known. For n large it is shown that this expected number in −∞,∞ is asymptotic to 2/π logn. In this paper, we show that this asymptotic value increases significantly to √ n 1 when we consider a polynomial in the form a0 ( n 0 ...

One of the main results of this paper is an upper bound for the total number of real isolated zeros of complete Abelian integrals, exponential in the degree of the form (Theorem 1 below). This result improves a previously obtained in [IY1] double exponential estimate for the number of real isolated zeros on a positive distance from the singular locus. In fact, the theorem on zeros of Abelian in...

We examine the behaviour of the zeros of the real and imaginary parts of ξ(s) on the vertical line Rs = 1/2 + λ, for λ 6= 0. This can be rephrased in terms of studying the zeros of families of entire functions A(s) = 1 2 (ξ(s+λ)+ξ(s−λ)) and B(s) = 1 2i (ξ(s+λ)−ξ(s−λ)). We will prove some unconditional analogues of results appearing in [3], specifically that the normalized spacings of the zeros ...

Abstract. Littlewood in his 1968 monograph “Some Problems in Real and Complex Analysis” [9, problem 22] poses the following research problem, which appears to still be open: “If the nm are integral and all different, what is the lower bound on the number of real zeros of PN m=1 cos(nmθ)? Possibly N − 1, or not much less.” Here real zeros are counted in a period. In fact no progress appears to h...

‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $alpha$‎, ‎let $D_{alpha}p(z)=np(z)+(alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| leq k, (kleq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$alphainmathbb{C}$ with $|alpha|geq k$‎, ‎begin{align*}‎ ‎max_{|z|=...