Related papers: A remark on an inequality for the prime counting f…
In the article we establish the Hardy-Littlewood inequality $ \pi (x + y) \leq \pi (x) + \pi (y) $. We also prove that the naturally ordered primes $p_1=2,p_2=3,p_3=5,p_4=7,\dots$ satisfy the inequality $ p_ {a + b}> p_a + p_b $ for all $a,…
An improved estimate is given for $|\theta(x) -x|$, where $\theta(x) = \sum_{p\leq x} \log p$. Three applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, and the…
We obtain an interesting inequalities for uniformly continuous functions in the normed spaces: $\|f(x)\|\leq a\|x\|+b$ for some $a,b> 0$.
It is proved that one cannot approximate stably the first derivative of a smooth function given noisy values of this function and a bound on this function and its first derivative. Such an approximation is shown to be possible if an a…
Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every…
In this note we study the Petty projection of a log-concave function, which has been recently introduced in [9]. Moreover, we present some new inequalities involving this new notion, partly complementing and correcting some results from…
In this note we prove an inequality involving primes and the product of consecutive primes.
Let $\theta(x) = \sum_{p\leq x} \log p$. We show that $\theta(x)<x$ for $2<x< 1.39\cdot 10^{17}$. We also show that there is an $x<\exp(727.951332668)$ for which $\theta(x) >x.$
We denote by $\pi\left( x\right) $ the usual prime counting function and let $li\left( x\right) $ the logarithmic integral of $x$. In 1966, R.S. Lehman came up with a new approach and an effective method for finding an upper bound where it…
An interesting episode in the history of the prime number theorem concerns a formula proposed by Legendre for counting the primes below a given bound. We point out that arithmetic bias likely played an important role in arriving at that…
Simple inequalities for some integrals involving the modified Struve function of the first kind $\mathbf{L}_{\nu}(x)$ are established. In most cases, these inequalities have best possible constant. We also deduce a tight double inequality,…
Rosser and Schoenfeld remarked that the product $\prod_{p\leq x}(1-1/p)^{-1}$ exceeds $e^{\gamma} \log x$ for all $2\leq x\leq 10^8$, and raised the question whether the difference changes sign infinitely often. This was confirmed in a…
In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $t_{\mu,\nu}(x)$. In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality,…
Log-concave distributions include some important distributions such as normal distribution, exponential distribution and so on. In this note, we show inequalities between two Lp-norms for log-concave distributions on the Euclidean space.…
Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular,…
We introduce approximation functions of $li(x)$ for all $x\ge e$: (1) $\displaystyle li_{\underline{\omega},\alpha}(x) = \frac{x}{\log(x)}\left( \alpha\frac{\underline{m}!}{\log^{\underline{m}}(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…
In this paper, we show that if $(U_n)_{n\ge 1}$ is any nondegenerate linearly recurrent sequence of integers whose general term is up to sign not a polynomial in $n$, then the inequality $\phi(|U_n|)\ge |U_{\phi(n)}|$ holds on a set of…
For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.
This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form $p=8k+1$, receives attention.