19 problems

Let $\Omega(n)$ denote the number of prime factors of $n$, counted with repetition. For $k\geq 1$ and $y\geq 1$, define the sifted Erdős sum … where the sum is over integers having…

Three prime squares conjecture. Every such $n$ can be represented as

Primality conjecture for the almost-prime Mill function. It is conjectured that $f(n)$ is prime for all $n\geq 1$. The paper proves that $f(n)$ has at most two prime factors counti…

Shanks' higher-order bias conjecture. The average of $\tau^{(a)}(n)$ is $(-1)^a$. This extends Chebyshev's bias to integers with a prescribed number of prime factors; the supplied…

Let $k\geq 1$ be odd, and let $N$ be a sufficiently large even number. Prime-power plus almost-prime conjecture. Then $N$ can be expressed as the sum of a $k$th power of a prime an…

Let $\widetilde{M}(k)$ denote the parameter in Theorem. Conjecture on $\widetilde{M}(k)$. One can take $\widetilde{M}(k)=k$ for all $k\geq 1$ in Theorem. This would improve the pap…

Let $\mathbb{P}_\ell$ denote the set of integers with the relevant almost-prime parameter $\ell$, let $\1_{\mathbb{P}_\ell}$ be its indicator, and let $\overline{\pi}_\ell(N)$ deno…

Let $x$ be sufficiently large, let $r$ be a sufficiently large integer, and define … where $P_s$ denotes an integer with at most $s$ prime factors counted with multiplicity. Wu-typ…

Asymptotic growth conjecture. The constants satisfy

Erdős's primitive set conjecture. The maximum of $f(A)$, ranging over all primitive sets $A$ of positive integers, is attained by the set of primes $A=\mathbb{N}_1$.

Let $\delta_2$ be the constant appearing in lower-bound sieve results for almost primes: if a set has level of distribution $x^{\vartheta-\varepsilon}$ with…

Constancy conjecture for $\Omega_k$ and $\Omega_k^{\mathrm{ext}}$. The function $\Omega_k(\vartheta,\vartheta_0)$ is constant with respect to $\vartheta_0$. The same applies to…

Let $N>6$ be even, and let $\Omega(n)$ denote the number of prime factors of $n$, counted with multiplicity. Semiprime Goldbach conjecture. Every such $N$ can be written as … where…

Consider the equation … Here $c>1$, $N$ is an integer, $m_1$ and $m_2$ are natural numbers, and $mathcal P_2$ denotes the set of natural numbers having at most two prime factors co…

Let $\Or$ be a thin orbit of primitive integral Pythagorean triples ${\bx}=(x,y,z)$, without affine injections, and define … Here $\cP(k)$ denotes the integers having at most $k$ p…

Let $\Or$ be an orbit of primitive integral Pythagorean triples ${\bx}=(x,y,z)$, and define … Here $\cP(5)$ denotes the integers having at most five prime factors. Coordinate-produ…

Let $\Or$ be a thin orbit of primitive integral Pythagorean triples, and let $\G$ be the acting group. For a triple ${\bx}=(x,y,z)$, define … Here $\cP(3)$ denotes the integers hav…

Fix $k\geqslant 3$, and consider numbers composed of exactly $k$ prime factors, namely numbers $p_1^{a_1}\cdots p_j^{a_j}$ with $a_1+\cdots+a_j=k$, distributed in arithmetic progre…

Let $q$ be a modulus, and let $a$ and $b$ be reduced residue classes modulo $q$, with $a$ a quadratic non-residue and $b$ a quadratic residue. For the counting function of products…