Related papers: The leading root of the partial theta function

Recently Alan Sokal studied the leading root $x_0(q)$ of the partial theta function $\Theta_0(x,q)=\sum\limits_{n=0}^\infty x^nq^{\binom n2}$, considered as a formal power series. He proved that all the coefficients of…

The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ %(where $(q,x)\in {\bf C}^2$, $|q|<1$) defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. For $q\in (-1,0)$ the…

We prove new properties of the zero set of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in (-1,0)\cup (0,1)$, $x\in \mathbb{R}$. We show that for each $q\in (0,1)$, there exists a line Re$x=-a$,…

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $(q,z)\in \mathbb{C}^2$, $|q|<1$. We show that for any $0<\delta _0<\delta <1$, there exists $n_0\in \mathbb{N}$ such that for any $q$ with…

The series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for $q\in [0,1)$, $x\in \mathbb{R}$, and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. For $q$ taking one…

We consider the partial theta function, i.e. the sum of the bivariate series $\theta (q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$ for $q\in (0,1)$, $z\in \mathbb{C}$. We show that for any value of the parameter $q\in (0,1)$ all zeros of the…

The series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for $|q|<1$ and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. It is known that for $q$ taking one of the…

The bivariate series $\theta (q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$, $\theta (q,.)$ is an entire function. We show that for $|q|\leq 0.108$ the function $\theta (q,.)$ has no…

We consider the partial theta function $\theta (q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $x\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. We show that, for any fixed $q$, if $\zeta$ is a multiple…

The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the…

We prove that for $q\in (0,1)$, the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ has no zeros in the closed domain $\{ \{ |x|\leq 3\} \cap \{${\rm Re}$x\leq 0\} \cap \{ |${\rm Im}$x|\leq 3/\sqrt{2}\} \} \subset…

Suppose l=2m+1, m>0. We introduce m "theta-series", [1],...,[m], in Z/2[[x]]. It has been conjectured that the n for which the coefficient of x^n in 1/[i] is 1 form a set of density 0. This is probably always false, but in certain cases,…

We prove that 1) for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ belong to the set $\{$~Re\,$x\in (-5792.7,0),$~$|$Im\,$x|<132~\}$ $\cup$ $\{…

We consider the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $q\in (-1,0)\cup (0,1)$ and either $x\in \mathbb{R}$ or $x\in \mathbb{C}$. We prove that for $x\in \mathbb{R}$, in each of the two cases $q\in…

Let [\theta] denote the integer part and {\theta} the fractional part of the real number \theta. For \theta > 1 and {\theta^{1/n}} \neq 0, define M_{\theta}(n) = [1/{\theta^{1/n}}]. The arithmetic function M_{\theta}(n) is eventually…

In 1907 M.Petrovitch initiated the study of a class of entire functions all whose finite sections are real-rooted polynomials. An explicit description of this class in terms of the coefficients of a series is impossible since it is…

We prove that for $q\in (-1,0)\cup (0,1)$, the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ has no zeros in the closed unit disk.

In this short note, a general result concerning the positivity, under some conditions, of the coefficients of a power series is proved. This allows us to answer positively a question raised by Guo (2010) about the sign of the coefficients…

For the partial theta function $\theta (q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$, $q$, $z\in \mathbb{C}$, $|q|<1$, we prove that its zero set is connected. This set is smooth at every point $(q^{\flat},z^{\flat})$ such that $z^{\flat}$ is…

Let $(x_n)$ be a positive real sequence decreasing to $0$ such that the series $\sum_n x_n$ is divergent and $\liminf_{n} x_{n+1}/x_n>1/2$. We show that there exists a constant $\theta \in (0,1)$ such that, for each $\ell>0$, there is a…