In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a se...In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.展开更多
In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbun...In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbund the points on the ecliptic, where this velocity is equal to the average velocity of the Sun over all the ecliptic. For this purpose he used the idea of infinitely small arcs and their ratios in different points of the circle. The great scientist Leonard Euler （1707-1783） introduced in his works on spherical trigonometry the line-element ds of the surface of the sphere, i.e. the differential of the arc length. He constructed the spherical trigonometry as an inner geometry on the surface of the sphere. He replaced the trigonometry lines, which were in use befbre him, by trigonometric functions.展开更多
The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbe...The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.展开更多
The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century...The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century and, before him, J. Wallis considered something vaguely similar. However, in the nineteenth century, the number circle was for the most part abandoned—even if something similar is on occasion accepted in geometry, in the sense that space is circular. The design of the present paper is to present positive proof of the veracity of the number circle and therefore, at the same time, to falsify the number line. Verifying the number circle implies falsifying negative infinity and positive infinity—infinity instead being neither negative nor positive, just like 0. Part of said proof involves showing that infinity can be defined both as 1+1+1+1+1+1+... and as -1-1-1-1-1-... and that the following Equation applies: 1+1+1+1+1+1+...=-1-1-1-1-1-... The principal mathematical technique that will be used to provide said proof is introduced here for the first time. It is called the two dimensional infinite series. It is an infinite series of infinite series. Some additional observations regarding the geography of infinity will be made. A more detailed description of the geography of infinity will be reserved for other papers. The Equation is discussed in this paper only to the extent that the attention that has been paid to it has necessitated the construction of a theory of infinity that, upon closer inspection, makes the Equation more self-evident and intuitively apparent;a fuller discussion will take place in a later paper.展开更多
文摘In this paper, we calculate the absolute tensor square of the Dirichlet L-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet L-functions. This study is a generalization of Akatsuka’s theorem on the Riemann zeta function, and gives a proof of Kurokawa’s prediction proposed in 1992.
文摘In one of his astronomical works the prominent arabic medieval scientists Thabit ibn Qurra （836-901） studied the visible motion of the Sun and found the points, where its velocity is maximum or minimum. He also lbund the points on the ecliptic, where this velocity is equal to the average velocity of the Sun over all the ecliptic. For this purpose he used the idea of infinitely small arcs and their ratios in different points of the circle. The great scientist Leonard Euler （1707-1783） introduced in his works on spherical trigonometry the line-element ds of the surface of the sphere, i.e. the differential of the arc length. He constructed the spherical trigonometry as an inner geometry on the surface of the sphere. He replaced the trigonometry lines, which were in use befbre him, by trigonometric functions.
基金the Guangdong Provincial Natural Science Foundation (No.05005928)the National Natural Science Foundation (No.10671155) of P.R.China
文摘The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.
文摘The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century and, before him, J. Wallis considered something vaguely similar. However, in the nineteenth century, the number circle was for the most part abandoned—even if something similar is on occasion accepted in geometry, in the sense that space is circular. The design of the present paper is to present positive proof of the veracity of the number circle and therefore, at the same time, to falsify the number line. Verifying the number circle implies falsifying negative infinity and positive infinity—infinity instead being neither negative nor positive, just like 0. Part of said proof involves showing that infinity can be defined both as 1+1+1+1+1+1+... and as -1-1-1-1-1-... and that the following Equation applies: 1+1+1+1+1+1+...=-1-1-1-1-1-... The principal mathematical technique that will be used to provide said proof is introduced here for the first time. It is called the two dimensional infinite series. It is an infinite series of infinite series. Some additional observations regarding the geography of infinity will be made. A more detailed description of the geography of infinity will be reserved for other papers. The Equation is discussed in this paper only to the extent that the attention that has been paid to it has necessitated the construction of a theory of infinity that, upon closer inspection, makes the Equation more self-evident and intuitively apparent;a fuller discussion will take place in a later paper.
