Let $x>1$ and $\zeta$ denote the Riemann zeta function. Is the equality
$$\int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\zeta(s)x^{s}}{s(s+1)(s+2)\cdots (s+k)} \mathrm{d}s = 2\pi i$$ possible for any integer $k>1$ where $\sigma_{0}>1$ ?
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$\begingroup$ The left-hand side is equal to $\frac{2\pi i}{k!} \sum_{n \leq x} (x - n)^k$. $\endgroup$– Peter HumphriesCommented Feb 27, 2020 at 14:57
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$\begingroup$ @PeterHumphries, any proof or reference ? $\endgroup$– user152442Commented Feb 27, 2020 at 18:29
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$\begingroup$ Chapter 5 of Montgomery-Vaughan - the bit on Cesaro sums with $a_n = 1$ for all $n$, so that $\alpha(s) = \zeta(s)$. $\endgroup$– Peter HumphriesCommented Feb 27, 2020 at 18:36
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$\begingroup$ @Peter Humphreys, thanks, just checked MV. But it seems your summand should rather be $(1-n/x)^k$... $\endgroup$– user152442Commented Feb 27, 2020 at 19:08
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$\begingroup$ Whoops, yep, that's correct $\endgroup$– Peter HumphriesCommented Feb 27, 2020 at 19:09
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