论文标题
分析里曼的假设
Analyzing Riemann's hypothesis
论文作者
论文摘要
在本文中，我们对Riemann的假设进行了详细的分析，涉及分析扩展的Zeta函数的零。 We use the functional equation $ζ(s) = 2^{s}π^{s-1}\sin{(\displaystyle πs/2)}Γ(1-s)ζ(1-s)$ for complex numbers $s$ such that $0<{\rm Re(s)}<1$ and the reduction to the absurd method where we use an analytical study based on a complex function and its modulus as a两个真实变量的实际功能与深度数值分析相结合，表明Riemann Zeta函数的非平凡零的实际部分等于$ 1/2 $，对于我们的最佳资源。这是分两个步骤完成的。首先，我们表明如果我们假设$ s $的实际部分的值在$ 0 $和$ 1 $之间，但与到达零可能矛盾的$ 1/2 $之间的值之间会发生什么。其次，假设没有真正的值$ y $，则通过应用逻辑规则来否定量化符和相应的摩根定律，我们也到达了一个可见的矛盾，从而使$ζ\左（1/2 +yi \ right）= 0 $。最后，我们分析$ y \ in \ mathbb r $应满足哪些条件，以便$ζ（\ displayStyle 1/2 +yi）= 0 $。尽管这些结果在我们的数值计算中是有效的，但我们没有观察并预见任何更改的趋势。我们的发现为从新的数学角度评估Riemman假设的有效性开辟了道路。
In this paper we perform a detailed analysis of Riemann's hypothesis, dealing with the zeros of the analytically-extended zeta function. We use the functional equation $ζ(s) = 2^{s}π^{s-1}\sin{(\displaystyle πs/2)}Γ(1-s)ζ(1-s)$ for complex numbers $s$ such that $0<{\rm Re(s)}<1$ and the reduction to the absurd method where we use an analytical study based on a complex function and its modulus as a real function of two real variables in combination with a deep numerical analysis to show that the real part of the non-trivial zeros of the Riemann zeta function is equal to $1/2$ to the best of our resources. This is done in two steps. Firstly, we show what would happen if we assumed that the real part of $s$ has a value between $0$ and $1$ but different from $1/2$ arriving at a possible contradiction for the zeros. Secondly assuming that there is no real value $y$ such that $ζ\left(1/2 +yi \right)=0$ by applying the rules of logic to negate a quantifier and the corresponding Morgan's law we also arrive to a plausible contradiction. Finally, we analyze what conditions should be satisfied by $y \in \mathbb R$ such that $ζ(\displaystyle 1/2 +yi)=0$. While these results are valid to the best of our numerical calculations, we do not observe and foresee any tendency for a change. Our findings open the way towards assessing the validity of Riemman's hypothesis from a fresh and new mathematical perspective.
