not greater than $x$ times the average gap.

One line of research to understand and give evidence for the conjectures above is to produce bounds of the form $$\label{prob1}
N^*(T) \leq (1+c) N(T),$$ and $$\label{prob2}
N(x,T) \gg N(T),$$ with $c,x>0$ as small as possible, as $T\to\infty$. These two problems have been widely studied with several improvements being made over the years. One of the approaches is to use some suitable explicit formula (relating sums with integrals) with an auxiliary function $f$ in some class ${\mathcal{A}}$ and produce an inequality relating the quantity we are interested to bound with some functional ${\mathcal{Q}}(f)$ over ${\mathcal{A}}$. Minimizing (or maximizing) the functional over the class ${\mathcal{A}}$ would then produce the best bound one can possibly get with that specific approach. Nowadays, this idea is a standard technique in analytic number theory (introduced first by Beurling and Selberg) and the following are some references (clearly not a complete list) where the main approach is exactly that: Large sieve inequalities [@GV; @HV]; Erdös-Turán inequalities [@CV2; @Va]; Hilbert-type inequalities [@CL3; @CLV; @CV2; @GMK; @GV; @Va]; Tauberian theorems [@GV]; Bounds in the theory of the Riemann zeta-function and $L$-functions [@CC; @CChi; @CChiM; @CCLM; @CCM; @CCM2; @CF; @CS; @Chi; 