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Define the mean value for function $f$ as $\lim \limits_{x \to \infty} \frac{1}{x} \sum \limits_{n \leq x} f(n)$ if the limit exists denoted as $M_f$

Define the logarithmic value for function $f$ as $\lim \limits_{x \to \infty} \frac{1}{\ln x} \sum \limits_{n \leq x} \frac{f(n)}{n}$ if the limit exists denoted as $L_f$

Its easy to prove that if $M_f$ exist then so does $L_f$ and they are equal, but the other way around is not true in general.

Let $F(s) = \sum \limits_{n=1}^{\infty} f(n) n^{-s}$ for all $s>1$, and we are given that in addition $F(s)$ satisfy $F(s) = \frac{A}{s-1} +o(\frac{1}{s-1})$ for $s \to 1^{+}$ and $A$ is a constant.

In addition assume that $L_f = A$, and use this to prove that $M_f= A$ ?

Note : This question has been uploaded to MSE but without any answer.

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  • $\begingroup$ Is this a homework problem? (I don't know how to do it, but that very precise condition with no context about where it comes from leads me to ask.) $\endgroup$
    – LSpice
    Mar 28, 2020 at 17:00
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    $\begingroup$ Link to the post on Mathematics: Relaxation to Mean and Logarithmic values for arithmetic function. $\endgroup$
    – Martin Sleziak
    Mar 28, 2020 at 17:01
  • $\begingroup$ @LSpice not a homework, i im intersted in NT and this question shows up in one of the notes in the subject i am reading. $\endgroup$
    – Ahmad
    Mar 28, 2020 at 17:02
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    $\begingroup$ BTW if you created two different MO accounts by accident, you can find some instructions related to merging here. $\endgroup$
    – Martin Sleziak
    Mar 28, 2020 at 17:28
  • $\begingroup$ The linked question is essentially answered in comments. $\endgroup$
    – metamorphy
    Aug 24, 2020 at 13:10

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