Related papers: Small values of the Euler function and the Riemann…

The Riemann Hypothesis is a conjecture that all non-trivial zeros of Riemann Zeta function are located on the critical line in the complex plane. Hundreds of propositions in function theory and analytic number theory rely on this…

Let $\delta>0$ and $\sigma=\frac{1}{2}+\tfrac{\delta}{\log T}$. We prove that, for any $\alpha>0$ and $V\sim \alpha\log \log T$ as $T\to\infty$, $\frac{1}{T}\text{meas}\big\{t\in [T,2T]: \log|\zeta(\sigma+\rm{i} \tau)|>V\big\}\geq…

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $\sigma(n)<e^\gamma n\log\log n$ holds for every integer $n>5040$, where $\sigma(n)$ is the sum of divisors function, and $\gamma$ is the…

We investigate the behavior of the Euler products of the Riemann zeta function and Dirichlet L-functions on the critical line. A refined version of the Riemann hypothesis, which is named "the Deep Riemann Hypothesis" (DRH), is examined. We…

The Euler product formula relates Dirichlet $L(s,\chi)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re…

The aim of this article is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of $L$-functions, namely, non-principal Dirichlet and those based on cusp…

We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such…

Euler presents a third proof of the Fermat theorem, the one that lets us call it the Euler-Fermat theorem. This seems to be the proof that Euler likes best. He also proves that the smallest power x^n that, when divided by a numer N, prime…

We obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} \varphi \left( \left\lfloor{x/n}\right\rfloor\right)$, involving the Euler functions $\varphi$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the…

This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that…

This paper investigates the relationship between the Riemann hypothesis and the statement $\forall n, ~g(n) \le e^{\sqrt{p_n}}$, where $g(n)$ is the maximum order of an element of $S_n$, the symmetric group on $n$ elements, and $p_n$ is the…

In the present paper, we prove a stability theorem for the Kaehler Ricci flow near the infimum of the functional E_1 under the assumption that the initial metric has Ricci > -1 and |Riem| bounded. At present stage, our main theorem still…

The Cram\'er-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \cCramer \, \log^2 p_n$ for some constant $\cCramer \geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\sum_{n=1}^N g_n <…

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…

We prove some new lower bounds for the counting function $\mathcal N_{\mathcal C}(x)$ of the set of Nov\'ak-Carmichael numbers. Our estimates depend on the bounds for the number of shifted primes without large prime factors. In particular,…

We obtain global explicit numerical bounds, with best possible constants, for the differences $\frac{1}{n}\sum_{k\leq n}\omega(k)-\log\log n$ and$ \frac{1}{n}\sum_{k\leq n}\Omega(k)-\log\log n$, where $\omega(k)$ and $\Omega(k)$ refer to…

Euler's totient function, $\varphi(n)$, which counts how many of $0,1,\dots,n-1$ are coprime to $n$, has an explicit asymptotic lower bound of $n/\log \log n$, modulo some constant. In this note, we generalise $\varphi$; given an…

A folklore proof of Euclid's theorem on the infinitude of primes uses the Euler product and the irrationality of $\zeta(2) = \pi^2/6$. A quantified form of Euclid's Theorem is Bertrand's postulate $p_{n+1} < 2p_n$. By quantifying the…

Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The…

Starting from the Euler's identity, the author improved Riemann's results, discovered the relationship between the Riemann Zeta function and the prime function, and obtained two new corollaries based on Riemann hypothesis is tenable. From…