9 problems

Let $1/2<\sigma<1$ be fixed. Montgomery's lower bound concerns the maximum of $\log|\zeta(\sigma+it)|$ for $t\in[T,2T]$ as $T\to\infty$: … where $c(\sigma)>0$. Montgomery's conject…

Let $d>0$ be fixed and let $T\to\infty$. Large-value distribution conjecture. … This predicts the tail probability of large central values and, when combined with the persistence o…

Let $Z_X(s)$ denote the truncated zeta-function used in the paper, and let $A>0$ be fixed. Truncated-zeta maximum conjecture. If $2<X<\log^A T$, then, as $T\to\infty$, … The conjec…

Let $S_k(\Gamma_0(N))$ be the space of holomorphic cusp forms of weight $k$ and level $N$. For $f\in S_k(\Gamma_0(N))$, let $L(s,f)$ be its associated Dirichlet series; its conduct…

Let $T>0$ and let $\zeta(s)$ denote the Riemann zeta-function. Maximum-size conjecture. … This addresses the “1 or 2?” question for the maximum of the zeta-function, predicting tha…

Aistleitner–Mahatab–Munsch conjecture. After removing the $-\log_4T$ term, the resulting bound is conjectured to be best possible up to the constant in the $O(1)$ term. The long re…

Let $M_{Dir}$ be the matrix associated with Dirichlet polynomials $D(t)=\sum_{n=N}^{2N}b_n e^{i t\log n}$, with $|b_n|\leq1$, and suppose $N\leq T\leq N^{O(1)}$. Montgomery's…

Let $\zeta(s)$ denote the Riemann zeta function, let $T$ be large, and fix $\sigma\in(1/2,1)$. Define the maximum … The conjecture is that the true order of magnitude of…

Let $L(s)$ be an element of the Selberg class, let $\sigma\in(1/2,1)$, and suppose that $L$ satisfies assumptions of the kind appearing in the paper's two large-values theorems, in…