اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA function to calculate all relative prime numbers up to the product of the first n primes
83
0
0.0
(
0
)
تأليف
Matthias Schmitt
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove an isomorphism between the finite domain from 1 up to the product of the first n primes and the new defined set of prime modular numbers. This definition provides some insights about relative prime numbers. We provide an inverse function from the prime modular numbers into this finite domain. With this function we can calculate all numbers from 1 up to the product of the first n primes that are not divisible by the first n primes. This function provides a non sequential way for the calculation of prime numbers.
قيم البحث
اقرأ أيضاً
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blo
cks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameter-free non-Markovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy. Our model satisfies the prime number theorem as an emerging property and a refined version of Cramers conjecture about the statistics of gaps between consecutive primes that seems closer to reality than the original Cramers version. Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilistic models like ours can help to get deeper insights about primes and the complex architecture of natural numbers.
We present several formulae for the large $t$ asymptotics of the Riemann zeta function $zeta(s)$, $s=sigma+i t$, $0leq sigma leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Sieg
el. Using these formulae, we derive explicit representations for the sum $sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $Phi(u,v,beta)$, $uin mathbb{C}$, $vin mathbb{C}$, $beta in mathbb{R}$. Generalizing the methodology used in the study of $zeta(s)$, we derive asymptotic formulae for $Phi(u,v,beta)$.
We consider the summatory function of the number of prime factors for integers $leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg Martin conject
ured that the difference of the summatory functions should attain a constant sign for all sufficiently large $x$. In this paper, we provide strong evidence for Greg Martins conjecture. Moreover, we derive a general theorem for arithmetic functions from the Selberg class.
We prove independence of normality to different bases We show that the set of real numbers that are normal to some base is Sigma^0_4 complete in the Borel hierarchy of subsets of real numbers. This was an open problem, initiated by Alexander Kechris, and conjectured by Ditzen 20 years ago.
We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof
are a new Type II estimate and using this estimate by applying Harmans sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selbergs eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
