9 problems

Let $q>1$ be an integer, let $F$ be a finite set of primes that does not contain all the primes dividing $q$, and let $a$ be a positive integer. Let $(N_i)_{i\geq 0}$ satisfy…

Let $p_1,p_2,\dots$ be the primes and let $P_k=\prod_{i=1}^k p_i$. A pair of consecutive positive integers is represented as a product of the first $k$ primes if $n(n+1)=P_k$. Ruth…

Let $q>1$ be an integer, let $F=\{p_1,\dots,p_k\}$ be a finite set of primes that does not contain all the primes dividing $q$, and let $a$ be a positive integer. Say that an integ…

Let $n= p_1^{k_1}p_2^{k_2}\dots p_m^{k_m}>5040$, where the $p_i$ are distinct primes and the $k_i$ are positive integers. For an index $j$, form the integer…

Let $p_1,p_2,\ldots,p_m$ be the first $m$ distinct primes, and let $n=\prod_{j=1}^{m}p_j>5040$. Robin's inequality is the assertion that $\sigma(n)/n<e^\gamma\log\log n$, where…

Let $N$ be a sufficiently large even integer, let $\gamma$ denote the Euler–Mascheroni constant, and let $p$ range over primes. Sabihi's second conjecture. The asymptotic formula ……

Growth and bound conjecture. The quantity $k^*$ grows slower than the upper bound $b^{1.02041}$ or the asymptotic upper bound $b^{1+\epsilon}$, where $\epsilon\to0$ as $b\to\infty$…

Let $\chi$ be the quadratic Dirichlet character used to measure the bias, and let $\mathcal{L}_\chi$ denote the associated bias constant. Nonvanishing bias conjecture. The quantity…

Let $r(x)$ denote the ratio measuring the bias in the theorem between products $pq\leq x$ with prescribed quadratic-character values and the corresponding unrestricted products. Po…