zeta函数的Euler乘积表示的正则化和Birch-Swinnerton-Dyer猜想
Regularizations of the Euler product representation for zeta functions and the Birch--Swinnerton-Dyer conjecture
摘要：
考虑了zeta函数的欧拉积表示的一种变式正则化，本文主要应用于Riemann zeta函数的欧拉积表示。正则化本身与求和表达式的zeta函数相同，但不使用莫比乌斯函数使我们能够确认剩余项的有限行为，这意味着除了临界线之外没有零。同样的方法也可以应用于与椭圆曲线相关的$L$ -函数，并且我们可以处理与Birch-Swinnerton-Dyer猜想密切相关的临界带极点的Taylor展开。
英文摘要：
We consider a variant expression to regularize the Euler product representation of the zeta functions, where we mainly apply to that of the Riemann zeta function in this paper. The regularization itself is identical to that of the zeta function of the summation expression, but the non-use of the Möebius function enable us to confirm a finite behavior of residual terms which means an absence of zeros except for the critical line. Same technique can be applied to the $L$-function associated to the elliptic curve, and we can deal with the Taylor expansion at the pole in critical strip which is deeply related to the Birch--Swinnerton-Dyer conjecture.
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