Related papers: Improved estimate for the prime counting function …

We consider the summatory function of the number of prime factors for integers $\leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg…

Let $V(T)$ denote the number of sign changes in $\psi(x) - x$ for $x\in[1, T]$. We show that $\liminf_{\;T\rightarrow\infty} V(T)/\log T \geq \gamma_{1}/\pi + 1.867\cdot 10^{-30}$, where $\gamma_{1} = 14.13\ldots$ denotes the ordinate of…

Let $\pi S(t)$ denote the argument of the Riemann zeta-function, $\zeta(s)$, at the point $s=\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)| \leq \left(\tfrac{1}{4} + o(1) \right)\tfrac{\log…

The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $\pi(x)$ function and Riemann's hypothesis.

In this article, we give explicit bounds of order $\log t$ for $\sigma$ close to $1$, for two quantities: $|\zeta'(\sigma +it)/\zeta(\sigma +it)|$ and $|1/\zeta(\sigma +it)|$. We correct an error in the literature, and especially in the…

Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet $L$-functions is true, we then establish…

We analyze union-find using potential functions motivated by continuous algorithms, and give alternate proofs of the $O(\log\log{n})$, $O(\log^{*}n)$, $O(\log^{**}n)$, and $O(\alpha(n))$ amortized cost upper bounds. The proof of the…

We shall give an explicit formula for $\psi(x, q, a)$ with an error term of the form $C/\log^\alpha x$ under the condition that $q<\log^{\alpha_1} x$ is nonexceptional, for various values of $\alpha$ and $\alpha_1$. We shall also give an…

Let $f(n)$ denote the number of unordered factorizations of a positive integer $n$ into factors larger than $1$. We show that the number of distinct values of $f(n)$, less than or equal to $x$, is at most $\exp \left( C \sqrt{\frac{\log…

Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…

For $k\ge1$, let $R_k(x)$ denote the reciprocal sum up to $x$ of numbers with $k$ prime factors, counted with multiplicity. In prior work, the authors obtained estimates for $R_k(x)$, extending Mertens' second theorem, as well as a…

An algorithm is devised for computing $\Phi(n) = \phi(1) + \phi(2) + \cdots + \phi(n)$ in time $\widetilde{\Theta}(n^{2/3})$ and space $\widetilde{\Theta}(n^{1/3})$. The starting point is an existing algorithm based on the Dirichlet…

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists).…

We demonstrate the impact of a generic zero-free region and zero-density estimate on the error term in the prime number theorem. Consequently, we are able to improve upon previous work of Pintz and provide an essentially optimal error term…

Let q be an odd positive integer and P \in F2[z] be of order q and such that P(0) = 1. We denote by A = A(P) the unique set of positive integers satisfying \sum_{n=0}^\infty p(A, n) z^n \equiv P(z) (mod 2), where p(A,n) is the number of…

In this paper we give a new semiprimality test and we construct a new formula for $\pi ^{(2)}(N)$, the function that counts the number of semiprimes not exceeding a given number $N$. We also present new formulas to identify the $n^{th}$…

This article considers the error term of the primes counting function. It applies some recent results on the densities of prime numbers in short intervals to derive an improvement of the error term from subexponential size to fractional…

Given the first 20-100 coefficients of a typical generating function of the type that arises in many problems of statistical mechanics or enumerative combinatorics, we show that the method of differential approximants performs surprisingly…

Let $f(n)$ be a random completely multiplicative function such that $f(p) = \pm 1$ with probabilities $1/2$ independently at each prime. We study the conditional probability, given that $f(p) = 1$ for all $p < y$, that all partial sums of…

Given $\beta\in(0,1)$, we show the existence of a Beurling generalized number system whose integer counting satisfies $N(x) = ax + O\bigl(x\exp(-c\log^{\beta} x)\bigr)$ for some $a>0$ and $c>0$, and whose prime counting function satisfies…