4 problems

The converse conjecture. If $S(q)\neq\emptyset$, then there exists a prime $p\equiv1\pmod q$ such that $z(p)\mid\pi(q)$.

Let $p>2$ be prime, and let $A'$ and $B'$ be coprime integers. A Ramanujan-type congruence for Euler's totient function $cvarphi$ would have the form … for every $n\geq0$. The toti…

Let $k$ be an even positive integer, and define … Set … … where the sum is over all $j$ such that $j$ and $j+k$ have the same prime factors, the starred product is over primes…

Let $p$ be an odd prime. The comparison of $varphi(p-1)$ and $varphi(p+1)$ exhibits a bias when $p$ and $p+2$ are both prime, with the former inequality appearing more often. Totie…