Noticing the chazzan's mistake while davening

Question

If you are still davening but you notice that the chazzan made a mistake (yes/no mashiv haruach during the wrong season, for example), and nobody else notices, can you interrupt your davening to correct him? Is this a valid צורך התפלה? It doesn't affect your personal davening per se, but it does affect the davening of the whole tzibbur in a more general sense.

I'd like an answer that covers several different parts of davening, but particularly shemoneh esrei and אלקי נצור since those are the most common places to be during chazarat hashatz.

Also, is the halacha different for parts that don't ruin shemoneh esrei if they're skipped, like זכרנו?

I admit the whole situation doesn't say much for the shul: the people who are supposed be paying attention to חזרת הש"ץ aren't, and the person who isn't supposed to pay attention is.

(Not from personal experience, fortunately.)

Answer 1

Halachipedia says:

If you notice that someone made a mistake in Davening that would require him to go back to the beginning, for example, you heard someone miss Mashiv HaRuach (and Morid HaTal), one may signal with one’s hands in order to get his attention, if that’s unsuccessful, one should tell him after Davening. However, this leniency isn’t agreed upon. [note 50]

note 50 - Rabbi Shlomo Zalman Auerbach in Halichot Shlomo 20:6 rules that one may signal to someone who made a mistake in Davening that would cause him to repeat Shemonah Esrei. This is also the opinion of Dvar Meshulam (pg 120) based on Igrot Moshe 4:16 (below). However, Rav Elyashiv (quoted in Peninei Tefillah pg 85-6) holds that it’s forbidden to signal to someone else because his mistake has nothing to do with your davening unless the other person’s mistake bothers his Davening. [From Sh”t Igrot Moshe (4:16 pg 28) who writes that in a case where there was an announcement of the incorrect page number for congregants who wouldn’t know any better that one may in Shaat HaDachak announce the correct page number between Brachot in Shemonah Esrei because the Tzorech Tefillah of others is like the Tzorech Tefillah of oneself, it seems that Rav Moshe would agree to Rabbi Shlomo Zalman.]

While this seems to speak about an individual's private Shemonah Esrei, I suggest the principle can be extended to the Chazzan's repetition when he made an error for which he would have to repeat his repetition to get it right or where the error bothers your davenning.

I have seen my Rav (still in his private Shemonah Esrei) gesture to the Chazzan under such circumstances. (Shame on me for not doing it first.)

Please note that Halachipedia does require the Chazzan to repeat his repetition if he made a critical error:

If the Shaliach Tzibbur makes a mistake in Chazara and forgets to say an addition has the same halacha as in individual who forgets that addition (with one exception).