I've noticed that $f_p(n) := \sum_{k=1}^nk^p$ has something interesting going with its zeros: $$ \begin{align} p = 0: &&0 \\ p = 1: &&-1, 0 \\ p = 2: && -1, -\frac12,0 \\ p = 3: &&-1,-1,0,0 \end{align} $$ where in the latter multiplicity of zeros of $f_p$ is counted. For now I can say that $0$ is always a zero, all zeros are real and all are spread in $[-1, 0]$. Unfortunately, the latter condition is not satisfied already for $p = 4$, however the two zeros that fall out of this interval are still real and their sum is $-1$. Is there anything interesting known about zeros of $f_p$ in general?
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4If $p$ is odd larger than $1$, then $-1,-1,0,0$ are zeros. If $p$ is even, then $-1,-\frac 12,0$ are zeros. See here. – mathlove Apr 14 '22 at 17:03
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@mathlove thanks, saw that. Yet nothing close to some proper study of the behavior of zeros – SBF Apr 14 '22 at 19:25
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4What about this question? – Fabius Wiesner May 05 '22 at 07:48
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Related: https://math.stackexchange.com/q/3770159/259363 – Anthony May 07 '22 at 09:08
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What are the zeroes of $p = 4$? I have a feeling that Bernoulli numbers are somehow involved. – Yajat Shamji Feb 10 '23 at 11:43