Related papers: Chebyshev's bias and generalized Riemann hypothesi…

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality $\sigma(n):=\sum_{p|n}p<e^{\gamma} n \log \log n$ is satisfied for $n > 5040$, where $\gamma$ denotes the Euler-Mascheroni constant. We show…

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants c_n, n =…

We present the first applications of the recently established by us (arXiv:1304.7895; Ukrainian Math. J. - 2014. -66. - P. 371-383) generalized Li's criterion equivalent to the Riemann Hypothesis. This criterion is the statement that the…

In this article we provide new explicit Chebyshev's bounds for the prime counting function $\psi(x)$. The proof relies on two new arguments: smoothing the prime counting function which allows to generalize the previous approaches, and a new…

Assuming the Generalized Riemann Hypothesis(GRH), we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 3.072 times the average spacing.

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log…

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we…

Let $R(n) = \sum_{a+b=n} \Lambda(a)\Lambda(b)$, where $\Lambda(\cdot)$ is the von Mangoldt function. The function $R(n)$ is often studied in connection with Goldbach's conjecture. On the Riemann hypothesis (RH) it is known that $\sum_{n\leq…

We note that the inequalities $0.92 \frac{x}{\log(x)} <\pi(x)< 1.11 \frac{x}{\log(x)}$ do not hold for all $x\ge 30$, contrary to some references. These estimates on $\pi(x)$ came up recently in papers on algebraic number theory.

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. For…

The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi(x)$ having a general expression of the form, \begin{align*} P(\pi(x)) - \frac{e x}{\log x} Q(\pi(x/e)) +…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

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…

We improve the lower bound for $V(T)$, the number of sign changes of the error term $\psi(x)-x$ in the Prime Number Theorem in the interval $[1,T]$ for large $T$. We show that \[ \liminf_{T\to\infty}\frac{V(T)}{\log…

A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this paper we present a generalization of this result to multiple…

Let $\chi$ be a primitive Dirichlet character whose conductor $q$ is a prime number. For the certain averages of values of $\log |L(s, \chi)|$ in $q$-aspect at a fixed $s=\sigma>1/2$, under Generalized Riemann Hypothesis (GRH), we explain…

Robin's Inequality posits $G(n)<e^{\gamma}$ for $n>5040$. Robin also showed that if the Riemann Hypothesis (RH) is false, then $G(n)>e^{\gamma}\left(1+\displaystyle\frac{c}{(\log n)^{b}}\right)$ for infinitely many values of $n$. By…

Let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, $A$ has a surjective residual modulo $\ell$ Galois representation. For $t\in \mathbb{Z}$ and $x>0$, denote…

One aspect of Chebyshev's bias is the phenomenon that a prime number, $ q $, modulo another prime number, $ p$, experimentally seems to be slightly more likely to be a nonquadratic residue than a quadratic residue. We thought it would be…

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,…