有限域上一元多项式环中的Mertens第二定理的二重推广

Double Mertens’ Second Theorem in the Polynomial Ring over a Finite Field

摘要: 2014年，罗马尼亚数论学家Popa通过引入素数双曲线法得到Mertens第二定理的二重推广： ∑_pq≤x1/pq=(ln(lnx)+B)²-ln²2+∫₀₊₀^1/2ln(1-x)/x dx+(ln(lnx)/lnx),其中p，q是素数，B是Mertens常数。在本文中，我们类比Popa的方法，运用Dirichlet双曲线法在有限域上的一元多项式环中得到了Mertens第二定理的二重推广，同时类比Rosen关于代数数域中Mertens第二定理的证明方法，重新证明了有限域上的一元多项式环中的Mertens第二定理。

Abstract: In 2014, by using the prime hyperbola method Popa developed the following double Mertens estimation: ∑_pq≤x1/pq=(ln(lnx)+B)²-ln²2+∫₀₊₀^1/2ln(1-x)/x dx+(ln(lnx)/lnx),where p, q are prime numbers and B is Mertens constant. In this paper, by modifying the methods of Popa and based on Dirichlet’s hyperbola method, we generalize the Mertens’ second theorem to the double case on the polynomial ring over a finite field. During this approach, by modifying Rosen’s method on the proof of the Mertens’ second theorem in algebraic number field, we give a new proof of the Mertens’ second theorem on the polynomial ring over a finite field.