15 problems

Let $G(x)$ denote the number of primes in the set $\left\{\left\lfloor x/n\right\rfloor:1\le n\le x\right\}$. Suppose that $x=p_1p_2p_3$, where $p_1,p_2,p_3$ are primes satisfying…

Let $n\geq 1$ be an odd integer, and define … Suppose that $\delta:=d_n(\lambda)\geq 2$ for some $\lambda\geq 1$. Disjoint zero-block conjecture. Each such $\lambda$ corresponds to…

Let $f(n)$ denote the sum defined earlier in the paper. The limiting-values conjecture. As $n\to\infty$, … … … … The paper motivates these limits through numerical plots and says t…

Let $n$ be an odd squarefree positive integer, and let $h^*(-p)=h(-p)$ for $p\geq7$ and $h^*(-3)=1/3$. The odd-squarefree class-number conjecture. If $n\equiv1\pmod{4}$, then … If…

Let $p$ and $q$ be distinct primes with $p\equiv q\equiv3\pmod{4}$, let $\alpha,\beta\geq1$, and define … Set $h^*(-p)=h(-p)$ for $p\geq7$ and $h^*(-3)=1/3$. The same-congruence cl…

Let $p$ and $q$ be distinct primes, let $\alpha,\beta\geq 1$, and define … Also set $h^*(-p)=h(-p)$ for $p\geq 7$ and $h^*(-3)=1/3$. The mixed-congruence class-number conjecture. I…

Let $\alpha>0$ be irrational, let $f_\alpha$ be the function defined in the paper, and let $\varphi$ denote the golden ratio. The golden-ratio range conjecture. … The authors state…

Let $\alpha\in(0,1)$ be irrational, and let $f_\alpha$ be the function defined in the paper. The unit-interval range conjecture. … This is presented as the assertion that every irr…

Let $\alpha>0$ be irrational, and let $f_\alpha$ be the function defined in the paper. The positive irrational range conjecture. … Together with the paper's established upper bound…

For $b\in\mathbb{N}\setminus\{1\}$, let $a\in\{1,\ldots,b-1\}$ be relatively prime to $b$, and let $s\in\mathbb{N}$. For the function $f_\alpha$ appearing in the paper, the range c…

Let $0<\theta\leq 1$ be real, and let $\pi_{\theta}(x)$ denote the number of integers $n$ with $1\leq n\leq x^{\theta}$ such that $\left\lfloor x/n\right\rfloor$ is prime. For a gi…

Let $r\geq 2$ be an integer. Let $f$ be given by the paper's definition in, with $h$ satisfying $|h(d)|\ll d^\alpha$ for some $\alpha\in[0,\frac{1}{2r-1}]$, and let $C_f$ be the co…

Let $a_0=1$, $a_1=2$, and define $a_n=3a_{n-1}-a_{n-2}$. For ${\boldsymbol{i}}=(i_1,i_2)\in\langle a_n\rangle\times\langle a_{n+1}\rangle$, let $u(\alpha({\boldsymbol{i}}))$ be the…

Farhi's conjecture. For each integer $m\geq 3$, every natural number $n$ can be represented as

Let $p$ be a prime number and let $a$ be an integer satisfying $1<a<p$. Consider the quotient $\left\lfloor \frac{p}{a}\right\rfloor$ for $a\leq\left\lfloor\sqrt{p}\right\rfloor$.…