Related papers: Extreme values of the Dedekind $\Psi$ function

This research article provides an unconditional proof of an inequality proposed by Srinivasa Ramanujan involving the Prime Counting Function $\pi(x)$, \begin{align*} (\pi(x))^{2}<\frac{ex}{\log x}\pi\left(\frac{x}{e}\right) \end{align*} for…

A work by Nicolas has shown that if it can be proven that a certain inequality holds for all $n$, the Riemann hypothesis is true. This inequality is associated with the Mertens theorem, and hence the Euler totient at $\prod_{k=1}^n p_k$,…

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erd\H{o}s, Selfridge, and…

Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\…

Write $T(n)$ as the sum of the reciprocals of the primes which divide $n$. Write $H(n) = \prod_{p|n}p/(p-1)$ where the product is over the prime divisors of $n$. We prove new bounds for $T(n)$ and $H(n)$ in terms of the smallest prime…

We investigate Riemann's xi function $\xi(s):=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$ (here $\zeta(s)$ is the Riemann zeta function). The Riemann Hypothesis (RH) asserts that if $\xi(s)=0$, then…

We show that each rational number $r$, $0\le r<1$, occurs as the fractional part of a Dedekind sum $S(m,n)$.

Let phi(n) denote the Euler totient function. We study the analytic part associated with the summatory function of sigma_1(n) and obtain explicit bounds under the Riemann Hypothesis. In particular, we establish an upper bound of order…

For $a\in \Bbb Z$ and $b\in\Bbb N$, $(a,b)=1$, let $s(a,b)$ denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs $(a_i,b_i)$, $i\in\Bbb N$, with $b_i$…

In this paper we provide a new series representation for the values of Riemann zeta function at integer arguments, namely: $ \zeta(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}\Gamma(1-\omega_{m}n)...\Gamma(1-\omega_{m}^{m-1}n)}{n!n^m}$, where…

We prove that the Riemann hypothesis is equivalent to the condition $\int_{2}^x\left(\pi(t)-\text{li}(t)\right)\mathrm{d}t<0$ for all $x>2$. Here, $\pi(t)$ is the prime-counting function and $\text{li}(t)$ is the logarithmic integral. This…

We analyze algorithms for computing the $n$th prime $p_n$ and establish asymptotic bounds for several approaches. Using existing results on the complexity of evaluating the prime-counting function $\pi(x)$, we show that the binary search…

The results of the study provide guidelines for the development and applications of algorithms. When the number of steps for calculating an assumption tends to infinity, probability theory can be applied to predict whether the assumption…

Using Watt's mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\varepsilon}$ for sufficiently…

Atkin and Garvan introduced the functions $N_k(n)$ and $M_k(n)$, which denote the $k$-th moments of ranks and cranks in the theory of partitions. Let $e_{2r}(n)$ be the $n$-th Fourier coefficient of $E_{2r}(\tau)/\eta(\tau)$, where…

The following is proven using arguments that do not revolve around the Riemann Hypothesis or Sieve Theory. If $p_n$ is the $n^{\rm th}$ prime and $g_n=p_{n+1}-p_n$, then $g_n=O({p_n}^{2/3})$.

First idea is to compute a quantity like the angular momentum with respect to (0, 0), of an unitary mass of coordinates (<[Xi(s)], =[Xi(s)]) while =[s] is the time, and, <[s] = constant. If we impose that the derivative along <[s], at…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

Let $p_r(n)$ denote the number of $r$-component multipartitions of $n$, and let $S_{\gamma,\lambda}$ be the space spanned by $\eta(24z)^\gamma \phi(24z)$, where $\eta(z)$ is the Dedekind's eta function and $\phi(z)$ is a holomorphic modular…