Related papers: Sharper bounds for the error term in the Prime Num…

We note an interesting and under-expressed fact from Chebyshev's initial bounding for the prime counting function, $\pi(x) := \# \{p \leq x : p \text{ prime}\},$ based upon a selection of fixed coefficients $d\in D$ to show $\psi(x) \asymp…

We improve the error term in the van der Corput transform for exponential sums \sum_{a \le n \le b} g(n) exp(2\pi i f(n)). For many functions g and f, we can extract the next term in the asymptotic, showing that previous results, such as…

We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $\varphi(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.

The order and disorder of binary representations of the natural numbers < 2^8 is measured using the BiEntropy function. Significant differences are detected between the primes and the non primes. The BiEntropic prime density is shown to be…

We show for decreasing, positive approximation functions $\psi$ such that $\tau = \lim_{q \to \infty} \frac{\log \psi(q)}{\log q} < \frac{13 + \sqrt{73}}{8}$ and such that $q^2 \psi(q) \to 0$ that the set $\text{Exact}(\psi)$ of numbers…

In 1977 Montgomery and Vaughan gave tight bounds for exponential sums of the form $\sum_{n\leq x}f(n)e(n\alpha)$ where $f$ is a $1$-bounded multiplicative function and $\alpha\in\mathbb R$, close to the conjectured $\ll \frac{x}{\sqrt{q}}+…

Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $\vert M(x) \vert$ that $ \left\vert M(x) \right\vert \sim \left[\frac{1}{\pi \sqrt{\varepsilon}(x+\varepsilon)}\right]\sqrt{x}$…

In this paper we propose an algorithm that correctly discards a set of numbers (from a previously defined sieve) with an interval of integers. Leopoldo's Theorem states that the remaining integer numbers will generate and count the complete…

We study convex empirical risk minimization for high-dimensional inference in binary models. Our first result sharply predicts the statistical performance of such estimators in the linear asymptotic regime under isotropic Gaussian features.…

In this article, we obtain effective estimates for the error term $\Delta_{k}(x)$ for all integers $k \geq2$, and completely explicit estimates for integers $k \in [3,9]$. The explicit results improve the powers of $x$ appearing in the…

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} \Lambda(m) \mathrm{e}(\alpha…

We derive and implement a new way to find lower bounds on the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the previously best known bound to 1.80203. Our method adds new constraints to Smyth's…

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We show that under the assumption of an…

In this paper we discuss a method to express the Prime counting function as a "sum" over Non-trivial zeros of Riemann Zeta function, using techniques from Analytic Number Theory, also we apply our results to the sum over primes of any…

Let $\{p_n\}_{n\ge 1}$ be the sequence of primes and $\vartheta(x) = \sum_{p \leq x} \log p$, where $p$ runs over the primes not exceeding $x$, be the Chebyshev $\vartheta$-function. In this note we derive lower and upper bounds for…

We introduce a sieve for counting twin primes up to a given range. Our method depends on a parameter ${\lambda}_x$ and the estimation of the number of twin primes obtained as a result, is called a fundamental structure of the distribution…

We describe a rigorous implementation of the Lagarias and Odlyzko Analytic Method to evaluate the prime counting function and its use to compute unconditionally the number of primes less than $10^{24}$.

The results of the computer investigation of the sign changes of the difference between the number of twin primes $\pi_2(x)$ and the Hardy--Littlewood conjecture $c_2\Li_2(x)$ are reported. It turns out that $\pi_2(x) - c_2\Li_2(x)$ changes…

We consider suspension semiflows of an angle multiplying map on the circle and study the distributions of periods of their periodic orbits. Under generic conditions on the roof function, we give an asymptotic formula on the number $\pi(T)$…

We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of…