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We consider a variant expression to regularize the Euler productrepresentation of the zeta functions, where we mainly apply to that of theRiemann zeta function in this paper. The regularization itself is identical tothat of the zeta function of the summation expression, but the non-use of theM\"oebius function enable us to confirm a finite behavior of residual termswhich means an absence of zeros except for the critical line. Same techniquecan be applied to the $L$-function associated to the elliptic curve, and we candeal with the Taylor expansion at the pole in critical strip which is deeplyrelated to the Birch--Swinnerton-Dyer conjecture.

Regularizations of the Euler product representation for zeta functions and the Birch--Swinnerton-Dyer conjecture