A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppo...A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.展开更多
In this article we discuss the explicit solvability of both Schwarz boundary value problem and Riemann-Hilbert boundary value problem on a half hexagon in the complex plane. Schwarz-type and Pompeiu-type integrals are...In this article we discuss the explicit solvability of both Schwarz boundary value problem and Riemann-Hilbert boundary value problem on a half hexagon in the complex plane. Schwarz-type and Pompeiu-type integrals are obtained. The boundary behavior of these operators is discussed. Finally, we investigate the Schwarz problem and the Riemann-Hilbert problem for inhomogeneous Cauchy-Riemann equations.展开更多
The Riemann-Hilbert boundary value problem for the inhomogeneous Cauchy-Riemann equation in the polydomain is considered. The sufficient and necessary solvable conditions and integral expressions of solution for the a...The Riemann-Hilbert boundary value problem for the inhomogeneous Cauchy-Riemann equation in the polydomain is considered. The sufficient and necessary solvable conditions and integral expressions of solution for the above problem are given.展开更多
In this paper we obtain non-isotropic weighted Lp estimates with the boundary distance weight function for the-equation on piecewise smooth strictly pseudoconvex domains under a hypoth- esis of complex transversality ...In this paper we obtain non-isotropic weighted Lp estimates with the boundary distance weight function for the-equation on piecewise smooth strictly pseudoconvex domains under a hypoth- esis of complex transversality in Cn using the explicit formula of solutions by Berndtsson-Andersson.展开更多
This paper, we discuss the solutions＇ characterize of Cauchy-Riemann equation and the extension phenomenon of Hartogs in C^n and, a series of new extended results of the solutions for Cauchy-Riemann equations is obta...This paper, we discuss the solutions＇ characterize of Cauchy-Riemann equation and the extension phenomenon of Hartogs in C^n and, a series of new extended results of the solutions for Cauchy-Riemann equations is obtained by using the latest developments of the solutions＇ extension. Furthermore, the case of the extension＇s limitation for the solutions is also given.展开更多
文摘A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.
文摘In this article we discuss the explicit solvability of both Schwarz boundary value problem and Riemann-Hilbert boundary value problem on a half hexagon in the complex plane. Schwarz-type and Pompeiu-type integrals are obtained. The boundary behavior of these operators is discussed. Finally, we investigate the Schwarz problem and the Riemann-Hilbert problem for inhomogeneous Cauchy-Riemann equations.
文摘The Riemann-Hilbert boundary value problem for the inhomogeneous Cauchy-Riemann equation in the polydomain is considered. The sufficient and necessary solvable conditions and integral expressions of solution for the above problem are given.
基金supported by the Korea Research Foundation Grant funded by Korea Government(MOEHRD,Basic Research Promotion Fund)(Grant No.KRF-2005-070-C00007)
文摘In this paper we obtain non-isotropic weighted Lp estimates with the boundary distance weight function for the-equation on piecewise smooth strictly pseudoconvex domains under a hypoth- esis of complex transversality in Cn using the explicit formula of solutions by Berndtsson-Andersson.
基金Supported by the EDSF of Shandong Province(J04A11)
文摘This paper, we discuss the solutions＇ characterize of Cauchy-Riemann equation and the extension phenomenon of Hartogs in C^n and, a series of new extended results of the solutions for Cauchy-Riemann equations is obtained by using the latest developments of the solutions＇ extension. Furthermore, the case of the extension＇s limitation for the solutions is also given.
