Definite integral of $\zeta(s)$ over the critical strip - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:38:54Z http://mathoverflow.net/feeds/question/94068 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94068/definite-integral-of-zetas-over-the-critical-strip Definite integral of $\zeta(s)$ over the critical strip Agno 2012-04-14T21:55:06Z 2012-04-14T21:55:06Z <p>Take the following definite integral:</p> <p>$$f(s):=\int_s^{1-s} \zeta(x) \mathrm{d} x$$ </p> <p>with $s \in \mathbb{C}$, $s=\sigma \pm ti$, $0&lt;\sigma&lt;1$ and $t,\sigma \in \mathbb{R}$.</p> <p>The graph of $|f(s)|$ shows a monotonically increasing function for $\sigma=\frac12$ (as expected, it 'plateaus' exactly at the $\rho$s) and an apparently strictly increasing function when $\sigma\ne\frac12$. </p> <p>There is however a small 'dip' in the area $1 &lt; t &lt; 3$, that unexpectedly induces a zero at $\frac12 \pm 2.50056818181399528638615277529..i$. For $\sigma\ne\frac12$ there are no zeros.</p> <p>Is there anything known about this zero? Could it be proven that it only exists for $\sigma=\frac12$? </p> <p>Thanks!</p>