Is There a Way to Regularize Euler Products on Primes?

[mhill]

although is not valid in general (since an Euler product usually converges only when Re(s) >1)

\frac{ d \zeta(1/2)}{\zeta (1/2)}= -\sum_{p} log(p)(1-p^{1/2}

 

[yasiru89]

Well, this is simply by taking logarithms on either side of the Euler product representation to get,

\log{\zeta(s)} = -\sum_{p} \log{(1 - p^{-s})}

where p[/tex] is the set of primes.<br /> Differentiating then gives,<br /> <br /> \frac{\zeta&amp;#039;(s)}{\zeta(s)} = -\sum_{p} (p^{s} - 1)^{-1} \log{p}<br /> <br /> This gives the zeta-regularized sum (and hence product) on primes (which looks curious as it is, unless special values of s[/tex] are used), but generally its more convenient to consider,&lt;br /&gt; &lt;br /&gt; \prod_{n} \lambda_{n} = \exp{-\zeta_{\lambda}&amp;amp;#039;(0)}&lt;br /&gt; &lt;br /&gt; for a zeta function defined on a sequence (\lambda_{n})_{n \geq 1}[/tex].&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; If its a prime regularization you&amp;amp;#039;re after, look for this paper by Munoz Garcia and Perez Marco called &amp;amp;#039;Super Regularization of Infinite Products&amp;amp;#039;.&amp;lt;br /&amp;gt; &amp;lt;br /&amp;gt; Never mind, here&amp;amp;#039;s the link to the preprint pdf-&amp;lt;br /&amp;gt; http://inc.web.ihes.fr/prepub/PREPRINTS/M03/M03-52.pdf