Let $\alpha>0$ and $A>0$. Suppose that $$R(X)=\int_{1}^{X}L(t)t^{-\alpha}\,dt,\,\,X>1$$ satisfies $\lim_{X\to\infty}R(X)=-A$. Then $L(t)<0$ for all $t$ sufficiently large.

Is it true?

  • 1
    $\begingroup$ Hint: By writing $L(t)=t^{\alpha}f(t)$ (where $f(t):=t^{-\alpha}L(t)$) and noting that the sign of $L(t)$ is the same as that of $f(t)$, you can simply ask this question for $f(t)$ considering $\int_1^X f(t)\, dt$. Now, do you know any function $f(t)$ that is not negative for all sufficiently large $t$ but will still give a negative integral as $X\to \infty$? $\endgroup$ – Minus One-Twelfth Jun 4 at 18:47
  • $\begingroup$ Yes...something like $f(t)=-1$, for $1\leq t<2$ and $f(t)=1/t^2$, for $t\geq 2$. $\endgroup$ – GoldSoundz Jun 4 at 19:05

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