检索结果-南通市图书馆

The Symmetric Series of Multiples of Primes

维普期刊数据库博看期刊评论

在线全文

学校读者我要写书评

暂无评论

Advances in Pure Mathematics 2022年第3期12卷 160-177页

作者： Pal Doroszlai horacio keller Independent Researcher Kékkut Hungary

The union of the straight and over the point of reflection—reflected series of the arithmetic progression of primes results in the double density of occupation of integer positions. It is shown that the number of free positions left by the double density of occupation has a lower limit function, which is growing to infinity. The free positions represent equidistant primes to the point of reflection: in case the point of reflection is an even number, they satisfy Goldbach’s conjecture. The double density allows proving as well that at any distance from the origin large enough—the distance between primes is smaller, than the square root of the distance to the origin. Therefore, the series of primes represent a continuum and may be integrated. Furthermore, it allows proving that the largest gap between primes is growing to infinity with the distance and that the number of any two primes, with a given even number as the distance between them, is unlimited. Thus, the number of twin primes is unlimited as well.

关键词： Twin Primes Prime Gaps Goldbach’s Conjecture

The Primes and Their Subsets as Continuum like the Integers

在线全文

维普期刊数据库

学校读者我要写书评

暂无评论

Advances in Pure Mathematics 2023年第12期13卷 733-750页

作者： Pal Doroszlai horacio keller Independent Researcher Kékkut Hungary

The present paper gives the proof of the set of primes as a continuum. It starts with the density of the primes, and shortly recapitulates the prime-number-formula and the complete-prime-number-formula. Reflecting the series of the primes over any prime gives the double density of occupation of integer positions by the union of the series of multiples of the primes. The remaining free positions render it possible to prove Goldbach’s conjecture and the set of primes as a continuum. The theoretical evaluation is followed in annexes by numerical evaluation, demonstrating the theoretical results. The numerical evaluation results in different constants and relations, which represent inherent properties of the set of primes.

关键词： Prime-Number-Formula Complete-Prime-Number-Formula Goldbach’s Conjecture Primes as Continuum

The Integral of the Inverse of the Primes

在线全文

维普期刊数据库

学校读者我要写书评

暂无评论

Advances in Pure Mathematics 2023年第5期13卷 250-266页

作者： Pal Doroszlai horacio keller Independent Researcher Kékkut Hungary

The present paper gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. First it starts with the density of the primes, shortly recapitulates the prime-number-formula and the complete-prime-number-formula, the proof of the set of primes as continuum. The theoretical evaluation is followed in annexes by numerical evaluation of the theoretical results and of different constants, which represent inherent properties of the set of primes.

关键词： Density of Primes Prime-Number-Formula Complete-Prime-Number-Formula Goldbach’s Conjecture Twin Primes K-Tuples of Primes Primes as Continuum

The Infinite Polynomial Products of the Gamma and Zeta Functions

在线全文

维普期刊数据库

学校读者我要写书评

暂无评论

Advances in Pure Mathematics 2022年第6期12卷 451-464页

作者： Pál Doroszlai horacio keller Independent Researcher Kékkút Hungary

Starting with the binomial coefficient and using its infinite product representation, the infinite product representation of the gamma function and of the zeta function are composed of an exponential and of a trigonometric component and proved. It is proved, that all these components define imaginary roots on the critical line, if written in the form as they are in the functional equation of the zeta function.

关键词： Gamma Function Zeta Function Critical Line

Erratum to “The Symmetric Series of Multiples of Primes” [Advances in Pure Mathematics, 12 (2022) 160-177]

在线全文

维普期刊数据库

学校读者我要写书评

暂无评论

Advances in Pure Mathematics 2022年第12期12卷 742-743页

作者： Pal Doroszlai horacio keller Independent Researcher Kékkut Hungary

The original online version of this article (Doroszlai, P. and keller, H. (2022) The Symmetric Series of Multiples of Primes, Advances in Pure Mathematics, Vol. 12, 160-177. http://***/10.4236/apm.2022.123014) unfortunately contains some small mistakes. The author discovered the possible problem in Equations (1.6), (1.7) and in the last equation of the proof of lemma 2.3 and the numbering of the chapters 3 and 4 should be changed. To fix the problem, the author wishes to change these equations and titles.

关键词： Erratum

维普期刊数据库博看期刊评论

在线全文

学校读者我要写书评

暂无评论

The Number of Primes

Advances in Pure Mathematics 2022年第2期12卷 81-95页

作者： Pál Doroszlai horacio keller Independent Researcher Kékkút Hungary

It is known that the prime-number-formula at any distance from the origin has a systematic error. It is shown that this error is proportional to the square of the number of primes present up to the square root of the distance. The proposed completion of the prime-number-formula in the present paper eliminates this systematic error. This is achieved by using a quickly converging recursive formula. The remaining error is reduced to a symmetric dispersion of the effective number of primes around the completed prime-number-formula. The standard deviation of the symmetric dispersion at any distance is proportional to the number of primes present up to the square root of the distance. Therefore, the absolute value of the dispersion, relative to the number of primes is approaching zero and the number of primes resulting from the prime-number-formula represents the low limit of the number of primes at any distance.

关键词： Prime-Number-Formula Prime Gaps

The Exponential Function as Split Infinite Product

在线全文

维普期刊数据库

学校读者我要写书评

暂无评论

Advances in Pure Mathematics 2022年第4期12卷 308-331页

作者： Pál Doroszlai horacio keller Independent Researcher Kékkut Hungary

It is shown that any polynomial written as an infinite product with all positive real roots may be split in two steps into the product of four infinite polynomials: two with all imaginary and two with all real roots. Equations between such infinite products define adjoint infinite polynomials with roots on the adjoint roots (real and imaginary). It is shown that the shifting of the coordinates to a parallel line of one of the adjoint axes does not influence the relative placement of the roots: they are shifted to the parallel line. General relations between original and adjoint polynomials are evaluated. These relations are generalized representations of the relations of Euler and Pythagoras in form of infinite polynomial products. They are inherent properties of split polynomial products. If the shifting of the coordinate system corresponds to the shifting of the imaginary axes to the critical line, then the relations of Euler take the form corresponding to their occurrence in the functional equation of the Riemann zeta function: the roots on the imaginary axes are all shifted to the critical line. Since it is known that the gamma and the zeta functions may be written as composed functions with exponential and trigonometric parts, this opens the possibility to prove the placement of the zeta function on the critical line.

关键词： Infinite Polynomials Shifting to the Critical Line Zeta Function