zeta函数的正则化欧拉积以及Birch和Swinnerton-Dyer和Beilinson猜想
Regularized Euler product for the zeta function and the Birch and Swinnerton-Dyer and the Beilinson conjecture
摘要：
我们提出了另一个表达式来正则化黎曼zeta函数的欧拉乘积表示。%，在本文中。这个表达式本身与通常的欧拉乘积即无穷乘积本质上是相同的，但我们定义了一个新的表达式作为由通常的欧拉乘积导出的某些项的乘积的极限。我们还提到伯努利数和$P(z)$之间的关系，它是逆素数的$z$幂的无限和。当我们将相同的技术应用于与椭圆曲线相关的$L$函数时，即使在临界带中，我们也可以评估该函数的泰勒展开式的能力，这与Birch和Swinnerton-Dyer猜想和Beilinson猜想等问题密切相关。
英文摘要：
We present another expression to regularize the Euler product representation of the Riemann zeta function. % in this paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a new one as the limit of the product of some terms derived from the usual Euler product. We also refer to the relation between the Bernoulli number and $P(z)$, which is an infinite summation of a $z$ power of the inverse primes. When we apply the same technique to the $L$-function associated to an elliptic curve, we can evaluate the power of the Taylor expansion for the function even in the critical strip, which is deeply related to problems known as the Birch and Swinnerton-Dyer conjecture and the Beilinson conjecture.
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