18 problems

Ivić's mean-square conjecture.

Maximal-projection conjecture. One has

Mean-value formula conjecture. The mean value is equal to

Let $S^{(1)}(x)$, $S^{(2)}(x)$, $C^{(1)}(x)$, and $C^{(2)}(x)$ be the subgroup-counting functions collected in $\mathcal{F}[x]$, and let … where $f(x)\in\mathcal{F}[x]$. Mean-squar…

Let … be the twisted mean-square asymptotic associated with an arbitrary Dirichlet polynomial of length $T^\theta$, where $a_n$ are arbitrary coefficients and $\theta$ denotes the…

Let $x$ be sufficiently large. Consider a real-valued completely multiplicative function $f$ satisfying $|f(n)|\leq 1$ for every positive integer $n$, and suppose that $f(p)=0$ for…

Square mean Lindelöf hypothesis. The value $\eta=0$ is admissible.

Koyama's mean Lindelöf hypothesis. The displayed bound should hold.

Let … where $\sigma\geq 1/2$ and $T\geq 2$. At $\sigma=1/2$, write … where $\gamma$ is the Euler--Mascheroni constant and $E(T)$ is the error term. Quarter-exponent error-term conj…

Let $\mathcal{I}$ and $\mathcal{J}$ be finite multisets of complex shifts, and let $\mathcal{M}_{0,\mathcal{I},\mathcal{J};\omega}(K)$ and…

Let $\mathcal{I}$ and $\mathcal{J}$ be finite multisets of complex shifts, with the shifted divisor coefficients and mean value $\mathscr{D}_{\mathcal{I},\mathcal{J};\omega}(K)$ de…

Shifted mean-square conjecture. One may conjecture that

Let $X$ be sufficiently large, and let ${\mathbf f}=(f_1,\ldots,f_t)$ be a well-conditioned $t$-tuple of polynomials with integral coefficients, where $\deg(f_j)=k_j$. Define … whe…

Let $I$ be the smoothed mean square … where $\phi$ is smooth and supported in $[1,2]$, and … For $\theta<1$, the Balasubramanian–Conrey–Heath-Brown formula … should remain valid. T…

Let $R(T)$ be defined by … Here $E^*(T)$ is the modified error term whose integral defines $R(T)$, and $\varepsilon>0$ is arbitrary. The conjecture on $R(T)$. One may conjecture th…

Let $a_n(k)$ denote the $k$th coefficient of the $n$th cyclotomic polynomial. Write $\mathcal A(k)$ and $\mathcal A_0(k)$ for the value sets defined in the paper, let…

Let $a_n(k)$ be the $k$th coefficient of the $n$th cyclotomic polynomial, let $M$ denote the mean value over $n$, and define $e_k$ by … For $k\geq1$, Möller's conjecture. One has ……

Conjecture. The short difference should satisfy