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Let $\xi_0, \xi_1, \dots$ be i.i.d. random variables with zero mean and unit variance. We study the following four families of random analytic functions: $\sum_{k=0}^n \sqrt{\binom nk} \xi_k z^k$ (spherical polynomials), $\sum_{k=0}^\infty \sqrt{\frac{n^k}{k!}} \xi_k z^k$ (flat random analytic function), $\sum_{k=0}^\infty \sqrt{\binom {n+k-1} k} \xi_k z^k$ (hyperbolic random analytic functions), $\sum_{k=0}^n \sqrt{\frac{n^k}{k!}} \xi_k z^k$ (Weyl polynomials). We compute explicitly the limiting mean density of real zeroes of these random functions. More precisely, we provide a formula for $\lim_{n\to\infty} n^{-1/2} \mathbb{E}N_n[a,b]$, where $N_n[a, b]$ is the number of zeroes in the interval $[a,b]$., Comment: 26 pages, 1 figure

Let $\xi_0,\xi_1,\ldots$ be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form $f(z)=\sum_{k=0}^\infty \xi_k c_k z^k$, where $c_0,c_1,\ldots$ is a real sequence such that $c_n^2$ is regularly varying with index $\gamma-1$, where $\gamma>0$. We prove that $\mathbb{E} N[0,1-\epsilon] \sim \frac{\sqrt{\gamma}}{2\pi} |\log \epsilon|$ as $\epsilon \downarrow 0$, where $N[0,r]$ denotes the number of real zeroes of $f$ in the interval $[0,r]$., Comment: 36 pages

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $$ X_n(t)=u+\frac{1}{\sqrt{n}}\sum_{k=1}^n (A_k\cos(kt)+B_k\sin(kt)), \quad t\in [0,2\pi],\quad u\in\mathbb{R} $$ whose coefficients $A_k, B_k$, $k\in\mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a, b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]\subseteq [0,2\pi]$, we prove that $$ \lim_{n\rightarrow\infty} \frac{\mathbb{E} N_n[a,b]}{n}=\frac{b-a}{\pi\sqrt{3}} e^{-\frac{u^2}{2}}. $$, Comment: 15 pages