About the Prime-number Theorem and the Zeros of the ζ-function

Abstract

The fundamental theorem of number theory, proved essentially by Euclid, states that every natural number can be decomposed in only one manner into a product of powers of different primes. We can therefore write, for Re (s) > 1

$$\zeta (s)\, = \,\sum\limits_{n = 1}^\infty {{1 \over {{n^s}}}} \, = \,\mathop {II}\limits_p (1 + {1 \over {{p^s}}} + {1 \over {{p^{2s}}}} + \cdot \cdot \cdot ) = \mathop {II}\limits_p {1 \over {1 - {p^{ - s}}}},$$

(44.1)

where the product is extended over all prime numbers þ. This product expansion was used already by Euler for special values of s. The absolute convergence of the product

$$\mathop {II}\limits_{} (1 - {1 \over {{p^s}}})$$

for Re (s) > 1 is readily seen since

$$\sum\limits_p {{1 \over {{p^\sigma }}} < \sum {{1 \over {{n^\sigma }}}} } ,\sigma > 1.$$

Since now in (44.1) all factors are different from 0 in the half-plane Re(s) > 1, the absolute convergence entails

$$\zeta (s)\, \ne 0$$

for Re(s) > 1.