Definition: Let $p_i$ be the $i$-th prime, $\pi(p_i)=i$ the prime counting function, and $g_i=p_{i+1}-p_i$ the prime gap. Observation - "Beta Density Law": For all primes $p_i$ with $i \leq 168$, i.e. $p_i \leq 997$, the gap satisfies: $$g_i \approx 0.065 \cdot \frac{p_i}{\pi(p_i)}$$ Note on...

I am studying a family of non‑homogeneous linear complex differential equations and encountered the following limit. I would like an explicit counterexample, if one exists. We consider $\eta \in L^\infty([1,+\infty))$ satisfying the following hypothesis $(H)$: $$ \exists \rho_\eta \in...

In my research project I am looking at a lower bound for Kloosterman sums, which are sums of roots of unity. The best known lower bound for a sum of $k$ $N$th roots of unity is $k^{-N}$, which comes from a simple algebraic number theory argument. In a 1986 paper, "How Small Can a Sum of Roots of...