Zero distribution of some difference polynomials

In this paper,suppose that a,c∈C{0},c_(j)∈C(j=1,2,···,n) are not all zeros and n≥2,and f (z) is a finite order transcendental entire function with Borel finite exceptional value or with infinitely many multiple zeros,the zero distribution of difference polynomials of f (z+c)-af^(n)(z) and f (z)f (z+c_1)···f (z+c_n) are investigated.A number of examples are also presented to show that our results are best possible in a certain sense.

THE DISTRIBUTION OF ZEROES OF SOLUTIONSOF NEUTRAL EQUATIONS

The purpose of ibis paper is to study the distribution of zerocs of solutions of theneutral delay differential equations. An estimate is estublished for the distween betweenadjacenlt zeroes of the solutions of such equations under less restritive hypotheses on the variable coefficients.The results obtained improve and extend scme known resultsin the literature.

Distribution of Zeros of Solutions of Advanced Differential Equations with One Advanced Variable

The distribution of zeros of solutions of the advanced differential equations with an advanced variable x′（t）-P（t）x（τ（t））=0,t≥t0 is studied, where P（t）∈C（[t0,∞）,R^＋）,τ：[t0,∞）→R^＋ are continuously differentiable and strictly increasing,τ（t）≥t and limt→∞τ（t）=∞.The estimate for the distance between adjacent zeros of the oscillatory solution of the above equation is obtained.

Proof of Riemann Conjecture

Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t −iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product expression , where zj are all roots of ξ(z), including complex roots. He proposed Riemann conjecture (RC): All roots of ξ(z) are real. As the product expression can only be used as a tool of contradiction, we prove RC by contradiction. To avoid the zeros of ξ(1/2 + it), define a subset . We have basic estimate , on L (R). One can construct by all real roots tj of ξ(t). If ξ has no complex roots, then w(t) = G(s)ξ(s) for s = 1/2 + it. If the product expression has a complex root z'=t' −iα, where 0 a ≤ 1/2, R' = |z′| > 10, then ξ(z) has four complex roots ±(t′ ± iα), and should contain fourth order factor p(z), i.e. ξ(z) = w(z)p(z). But p(z) can not be contained in ξ(s), as we have on L(R) and p(t) ≥ 0.5(t/R)4 . As a result, we can rewrite ξ(t) = w(t)p(t) =G(s)ξ(s)p(t) on and get This contradicts the basic estimate. Therefore ξ(z) has no complex roots and RC holds.

THE DISTRIBUTION OF ZEROS OF SOLUTIONS OF NEUTRAL DIFFERENTIAL EQUATIONS WITH A VARIABLE DELAY

In this paper, the distribution of zeros of solutions of the first-order neutral differential equation with a variable delayis studied. The estimate for the distance between adjacent zeros of the oscillatory solution of the above equation is obtained.

The Proof of Riemann Hypothesis, the Key to the Door Is the Periodicity

The Riemann hypothesis is a well-known mathematical problem that has been in suspense for 162 years. Its difficulty lies in the fact that it is involved in an infinite integral which includes infinite series with complex variables. To detour this is in vain, since all the messages are hid in it. To unscramble them, there is a totally new idea, that is, the “periodicity”! By investigating the numerical approximate values of zero points, an explicit distribution law on the critical line was found. To accord with this, a periodic form for the real part of Xi function was constructed and rigidly proved. The Riemann hypothesis can be divided into three progressive propositions. The first proposition (the number of zero points in the critical strip satisfies a certain estimation) had been proved in 1905. The second proposition (the number of zero points on the critical line satisfies the same estimation as in the critical strip) is ever in suspense. It can be solved perfectly with the newly found “periodicity”. The third proposition (all the nontrivial zero points are on the critical line), that is, the Riemann hypothesis, is also true. The proof is a combination of the symmetry, monotonicity, periodicity of the Xi function and the extremum principle of the harmonic functions. It is the moment to draw full stop for this suspending problem.

Angular distribution in complex oscillation theory

Let f1 and f2 be two linearly independent solutions of the differential equation f" + Af =0,where A is an entire function.Set E-f1f2.In this paper,we shall study the angular distribution of E and establish a relation between zero accumulation rays and Borel directions of E.Consequently we can obtain some results in the complex differential equation by using known results in angular distribution theory of meromorphic functions.