arXiv subjects

Yuri Heymann

Publications and source records attributed to Yuri Heymann.

5 recordsLinked to original sources

arXiv2023

p-composites of the main sequence of odd numbers as building blocks for $\pi(x)$

The prime-counting function $\pi(x)$ which returns the number of primes smaller or equal to a given number is a topic of interest in number theory. An algorithm based on a cyclic group isomorphic to $Z/nZ$, the so-called $Z$-functions, was proposed in view to outperform its pieers. The approach suggests a time complexity $\mathcal{O}(x^{1/2})$ in agreement with optimality of a 2-D squared adaptive-recursive algorithm. The present work is a presentation of various approaches as ascending factorization, the main sequence of odd numbers and partial sequences, T-series, counting function of prime composites, $Z$-modular forms and combinatorial aspects.

math.GM↗

arXiv2021

The moment-generating function of the log-normal distribution, how zero-entropy principle unveils an asymmetry under the reciprocal of an action

The present manuscript is about application of It{\^o}'s calculus to the moment-generating function of the lognormal distribution. While Taylor expansion fails when applied to the moments of the lognormal due to divergence, various methods based on saddle-point approximation conjointly employed with integration methods have been proposed. By the Jensen's inequality, the MGF of the lognormal involves some convexity adjustment, which is one of the aspects under consideration thereof. A method based on zero-entropy principle is proposed part of this study, which deviations from the benchmark by infinitesimal epsilons is attributed to an asymmetry of the reciprocal. As applied to systems carrying vibrating variables, the partial offset by the reciprocal of an action, is a principle meant to explain a variety of phenomena in fields such as quantum physics.

math.GM↗

arXiv2020

An algorithm for the prime-counting function of primes larger than three

The prime-counting function $\pi(x)$ which computes the number of primes smaller or equal to a given real number has a long-standing interest in number theory. The present manuscript proposes a method to compute $\pi(x)$ with time complexity $\mathcal{O}(x^{1/2})$ without the need to introduce the non-trivial zeros of the Riemann zeta function.

math.GM↗

arXiv2020

A lower bound for the modulus of the Dirichlet eta function on a partition $\mathcal{P}$ from 2-D principal component analysis and transitive composition

The present manuscript aims to derive an expression for the lower bound of the modulus of the Dirichlet eta function on vertical lines $\Re(s)=\alpha$. The approach employs concepts of two-dimensional principal component analysis built on a parametric ellipse, to match the dimensionality of the complex plane. The one-sided lower bound $\forall s \in \mathbb{C}$ s.t. $\Re(s) \in \mathcal{P}$, $| \eta(s) | \geq \left| 1 - \frac{\sqrt{2}}{2^\alpha} \right|$, where $\eta$ is the Dirichlet eta function, is related with the Riemann hypothesis as $|\eta(s)| > 0$ for any $s \in \mathbb{C}$ s.t. $\Re(s) \in \mathcal{P}$, where $\mathcal{P}$ is a partition spanning one half of the critical strip depending upon a variable. We propose the composite lower bound $\forall s \in \, \mathbb{C}$ s.t. $\Re(s) \in \,]1/2,1[$, $|\eta(s)| \geq \text{Min}\left(1- \frac{\sqrt{2}}{2^{\alpha}},\frac{\sqrt{2}}{2^\alpha}-\frac{\sqrt{2}}{2}\right)$, resulting from transitive composition in $\eta(s) = \left(1-\frac{2}{2^s} \right) \zeta(s)$. As a founding principle, the solution space of the set of solutions referring to such $\mathcal{L}^2$-problem is a representation of the space spanned by explanatory variables satisfying its algebraic form.

math.GM↗

arXiv2018

An investigation of the non-trivial zeros of the Riemann zeta function

While many zeros of the Riemann zeta function are located on the critical line $\Re(s)=1/2$, the non-existence of zeros in the remaining part of the critical strip $\Re(s) \in \, ]0, 1[$ is the main scope to be proven for the Riemann hypothesis. The Riemann zeta functional leads to a relation between the zeros on either sides of the critical line. Given $s$ a complex number and $\bar{s}$ its complex conjugate, if $s$ is a zero of the Riemann zeta function in the critical strip $\Re(s) \in \, ]0, 1[$, then $\zeta(s) = \zeta(1-\bar{s})$, as a key proposition to prove the Riemann hypothesis.

math.GM↗