Given a real polynomial $f(x)=a_0x^d+\cdots+a_1 x$, I want to give a sharp estimate for $\sum_{n\le X} e(f(n))$. If $f(x)=ax$ is a linear function, we have $$\sum_{n\le X} e(\alpha n)\ll \min (X,\|\alpha\|^{-1});$$ If $f(x)\in\mathbb{Z}[x]$ where $p$ is a prime, using Weil's bound for...

Let $A\in GL(r,\mathbb{Z})$ be a positive definite matrix, $Q(x)=\dfrac{1}{2} x^t A x$ be the quadratic form associate to $A$, $Q^*(x)=\dfrac{1}{2} x^t A^{-1} x$ be the adjoint form of $Q(x)$ . Given $(c,d)=1,m\in\mathbb{Z}^r,$ we define $$G_m\left(\dfrac{d}{c}\right)=\sum_{h\,\text{mod}\,c}...

I am a curious outsider to mathematics and recently started reading about the Riemann Hypothesis. I am aware that many outsiders mistakenly believe they have solved the Riemann Hypothesis. I am not making such a claim. I am only trying to understand whether this perspective already exists in the...

In Vaughan's paper "A variance for k-free numbers in arithmetic progressions" https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024611505015352 he uses at one point a kind of dyadic decomposition for major arcs. From my understanding, I think he has $$\sum _{q\leq R}\int_{|\beta...

Title Analytic validity of mapping the regularized sum of natural numbers to the sum of primes via a scalar product projection Body In a recent preprint by Ezadiin Redwaan titled "A Scalar Product Approach to Strong Goldbach, Twin Primes, Polignac Conjectures And Geometric Unification of...

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...