16 problems

Berndt–Kim's sign conjecture. For sufficiently large $n$, the coefficients $a_n$ have the same sign.

For $q\in(-1,0)$, let $\theta(q,\cdot)$ be the partial theta function, let $x_1(q)$ denote its first positive zero, and consider a conjugate pair of zeros $a\pm ib$ with $a>0$, whe…

Let … be an entire function. The monotonicity conjecture states that if, for some $a_1>1$, membership of $F_{a_1}$ in the Laguerre–Pólya class is equivalent to the existence of … s…

Let $\theta(q,x)$ be the partial theta function. For $q\in(0,1)$, the zeros are complex-conjugation invariant, so nonreal zeros occur in complex conjugate pairs; for each…

Let $\theta(q,x)$ be the partial theta function, and let a negative real zero mean a real number $x<0$ satisfying $\theta(q,x)=0$. Lower-bound conjecture. Any negative real zero of…

Let $q\in(0,1)$, and let $\theta(q,x)$ be the partial theta function. For each $j\in\mathbb{N}$, let $\Gamma_j$ be the smooth curve formed by the graphs of the zeros $\xi_{2j-1}$ a…

Uniqueness conjecture. The spectral values $\tilde{q}_1$ and $v_{\pm}$ are the only spectral values of $\theta$ for

Let … where $\widetilde q\approx 0.3092493386$ is the positive threshold described in the source. Sokal's partial-theta problem. Is it true that all zeros of $\Theta_0(z;q)$ remain…

Let $\theta(q,.)$ be the partial theta function considered in the paper, and let $\bar q_k$ and $\bar y_k$ be the sequence of parameter values and corresponding double real zeros i…

Iteration-convergence conjecture. If $0<q\leq\widetilde q$, then $\{f_j(x)\}$ converges uniformly on $[-1,0]$, together with all derivatives, to

Critical-spectrum conjecture. The spectrum $\mathfrak S$ contains infinitely many positive values

Let $\widetilde{R}(x,y,q)$ be the rescaled three-variable Rogers–Ramanujan function, and let $\mathcal{S}_\alpha$ be the classes of formal power series defined by coefficientwise n…

Let $\widetilde{R}(x,y,q)$ be the rescaled three-variable Rogers–Ramanujan function, and for $\alpha\in\mathbb{R}\setminus\{0\}$ let $\mathcal{S}_\alpha$ denote the class of formal…

Let $F(x,y)=\sum_{n=0}^{\infty}\frac{x^n}{n!}y^{n(n-1)/2}$ be the deformed exponential function. For $\alpha\in\mathbb{R}\setminus\{0\}$, let $\mathcal{S}_\alpha$ be the class of f…

Euler-transform coefficient conjecture. The sequence $(c_m)_{m=1}^{\infty}$ is strictly positive, increasing, strictly convex, and satisfies $\Delta^k c\geq0$ for $k=3,4$; explicit…

Let $\widetilde{R}(x,y,q)$ be the rescaled three-variable Rogers–Ramanujan function, and write … where each $P_n(q)$ is a self-inversive polynomial in $q$ with integer coefficients…