Does the Euler product formula diverge for any zero of the Riemann zeta function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:34:49Z http://mathoverflow.net/feeds/question/63787 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63787/does-the-euler-product-formula-diverge-for-any-zero-of-the-riemann-zeta-function Does the Euler product formula diverge for any zero of the Riemann zeta function? Seongsoo Choi 2011-05-03T09:51:46Z 2011-05-05T07:50:00Z <p><strong>Simple question (but not for me):</strong> Does the Euler product formula diverge for any zero of the Riemann zeta function?</p> <p>The reason why I ask this is that I heard we should not use the Euler product instead of the Riemann zeta function for Re(s)=&lt;1 because it diverges on the critical strip, but I am not sure of that.</p> <p>According to my numerical calculation, it seems that it converges for the (known) zeta zeros.</p> <p><strong>Additional Question 1:</strong> Is it clear that the Euler product for a nontrivial zeta zero is either divergent or convergent?</p> <p><strong>Additional Question 2:</strong> If only one case is possible, which one is the right answer? Divergent or convergent?</p> http://mathoverflow.net/questions/63787/does-the-euler-product-formula-diverge-for-any-zero-of-the-riemann-zeta-function/63806#63806 Answer by GH for Does the Euler product formula diverge for any zero of the Riemann zeta function? GH 2011-05-03T14:07:45Z 2011-05-03T19:58:32Z <p>I am in a hurry now but let me tell what I think. I believe that in the critical strip and off the real axis $\prod_p (1-p^{-s})$ does not converge to any complex number (including zero). Using a similar idea as in my response to your related earlier question, this boils down to the fact that for any nonzero constants $c_1,\dots,c_n$ the sum $\sum_{m=1}^n c_m \sum_{p\in P}p^{-ms}$ oscillates wildly as $P\to\infty$. I dont's see this immediately, but I believe what happens is that the oscillation (or divergence) behavior of the inner sum depends heavily on $m$. More precisely, I believe that for $m=1$ you get a much wilder behavior than for the rest $m>1$. So altogether the above double sum inherits the behavior of $m=1$, namely the existence of very large partial sums for infinitely many $P$'s, and this prevents convergence or a tendency to pointing in special directions. I apologize if this is too vague, but certainly more than a comment.</p> <p><strong>EDIT 1:</strong> David Speyer showed that in the critical strip $\prod_p (1-p^{-s})$ does not converge to any <em>nonzero</em> complex number, see <a href="http://mathoverflow.net/questions/63714/is-the-euler-product-formula-always-divergent-for-0res1" rel="nofollow">here</a>. I believe that my approach above can also be made to work and yield more information. Perhaps the Riemann Hypothesis can be of great assistance here as $\sum_{p\in P}p^{-s}$ is very subtle. Note that for $\mathrm{Re}(s)=1$ and $s\neq 1$ the Euler product does converge to $\zeta(s)$, see Section 3.15 in Titchmarsh: The Theory of the Riemann Zeta-function.</p> <p><strong>EDIT 2:</strong> In my response to <a href="http://mathoverflow.net/questions/63714/is-the-euler-product-formula-always-divergent-for-0res1" rel="nofollow">this question</a>, I outline the proof that, assuming the Riemann Hypothesis, the partial products of $\prod_p (1-p^{-s})$ get arbitrary close to $0$ and $\infty$, at least for $\frac{1}{2}&lt;\mathrm{Re}(s)&lt;1$. I don't see any fundamental difficulty in extending this to $\mathrm{Re}(s)=\frac{1}{2}$.</p>