12 problems

For a given arithmetic function $f$, let $F_f(x)$ be the largest $k$ for which there exist integers $n$ with $n+k\leq x$ and … Here $\varphi$ denotes Euler's totient function. Erdő…

Shapiro's conjecture. There is at least one prime at each height; equivalently, $pi_n\geq 1$ for every height $n$.

For a nonnegative integer $s$, define … Here $\varphi_k(n)$ is the generalized totient function … and $\varphi_0(n)=\varphi(n)$ is Euler's totient function. Büyükaşık–Göral–Sertbaş…

Let $\Phi(L)=\sum_{n=1}^L\varphi(n)$ and define … The function $E(L)$ is the error term in the summatory totient function; the source records lower bounds for its oscillation. Mont…

Let $\Phi(L)=\sum_{n=1}^L\varphi(n)$ and define … Here $E(L)$ is the error term in the summatory totient function, whose size has been studied through both upper and lower bounds.…

Erdős's average-order conjecture. The average order is

Let $[t]$ denote the integral part of the real number $t$, and let $\varphi$ be Euler's totient function. As $x\rightarrow\infty$, the totient-sum conjecture. … This conjecture sha…

Let $\mathcal{V}=\{\phi(n):n\geqslant 1\}$ be the set of values of Euler's totient function, and let $\mathcal{D}$ be the set of positive integers that occur infinitely often as di…

Signed error-term independence conjecture. Theorem 3.1 should remain valid for these coefficients and dilates:

Let $A(m)=|\{n:\phi(n)=m\}|$ be the multiplicity of a totient $m$. Erdős's conjecture. Every real number $c_4<1$ is admissible as an exponent in the assertion that there are infini…

Let $\phi$ be Euler's totient function. For $x\geqslant 1$, let $V(x)$ be the number of totients at most $x$, and for an integer $k\geqslant 1$, let $V_k(x)$ be the number of totie…

Joint fiber conjecture. For every $k\geqslant 1$ and $l\geqslant 2$, there are integers $n$ such that