15 problems

Let … Here $q$ is a complex number with $|q|<1$, so the double Lambert series defines an analytic function in its domain of convergence. Andrews–Dixit–Schultz–Yee conjecture. The f…

Let $\Delta(z)$ be the Ramanujan delta cusp form, with Fourier expansion … For $z=x+iy$, consider the automorphic function $|\Delta(x+iy)|^2$ and its associated Lambert series…

Let $r$ be a positive integer. The notation $[q^n]A(q)$ denotes the coefficient of $q^n$ in the series $A(q)$. Amdeberhan–Andrews–Ballantine coefficient identity. … The identity wa…

Amdeberhan–Andrews–Ballantine conjecture.

Identity conjecture.

Let $r$ be a positive integer, and let $[q^k]F(q)$ denote the coefficient of $q^k$ in $F(q)$. Coefficient conjecture. Then … This is a proposed coefficient relation between two Lam…

Let $a$ be a positive integer, let $n$ be a positive integer, and write $[q^k]F(q)$ for the coefficient of $q^k$ in $F(q)$. Let … be the sum of the divisors of $n$. Coefficient con…

Zagier's conjecture. The function $c_0(y)$ has an oscillatory behaviour as $y\to0^+$.

A Lambert-series evaluation of an odd zeta value has an integer merit $a$ when the asymptotic ratio of successive terms is $\exp(-\pi\sqrt{a})$. The paper considers evaluations of…

For $k=6n+1$, define $q_3=\exp(-\pi\sqrt{3})$ and coefficients $E_{k,d}$ by … Define … where $e_n$ is the sequence specified by the displayed formula and recurrence in the source.…

Explicit generalized-factorization conjecture. If

Let $L(1,0,\gamma,\delta;q)$ denote the generalized Lambert series expansion and let $\gamma_k(n)$ be the arithmetic functions defined through the inverse matrix entries … Here…

Let $p(n)$ denote the partition function, let $\mu(n)$ denote the Möbius function, and let $s_{n,1}^{(-1)}$ denote the entries in the first column of the inverse matrix under consi…

Zero-magnitude conjecture. The zeros of $\widetilde{Q}_n(q)$, equivalently the reciprocal zeros of $Q_n(q)$, have maximum magnitude of a little over $1$, depending on $n$. The sour…

Chowla's conjecture. The values $f(x)$ and $g(x)$ are irrational at all such rational values of $x$.