
mahnen
to remind
To firmly remind someone of a payment that is due, a debt, or another obligation, often in written form.
Die Bibliothek hat mich gemahnt, die überfälligen Bücher zurückzugeben.
The library reminded me to return the overdue books.
to remind / warn
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to remind
To firmly remind someone of a payment that is due, a debt, or another obligation, often in written form.
Die Bibliothek hat mich gemahnt, die überfälligen Bücher zurückzugeben.
The library reminded me to return the overdue books.

to warn
To urgently call on someone to behave in a certain way or warn them about something, usually for moral or safety-related reasons.
Die Umweltschützer mahnen zu einem sparsameren Umgang mit Ressourcen.
The environmental activists warn that resources should be used more sparingly.

to commemorate
To serve as a reminder of a past, usually negative event.
Das Denkmal mahnt an die Opfer des Krieges.
The memorial commemorates the victims of the war.
'Mahnen' (to remind / warn) is a formal word that is often used in written language, especially in official or business contexts.
Das Unternehmen mahnt seine Kunden, die Rechnungen pünktlich zu bezahlen.
The company reminds its customers to pay their invoices on time.
The verb 'mahnen' (to remind / warn) is closely related to the noun 'die Mahnung' (reminder notice). A Mahnung is the written document you receive when you are formally reminded to pay or fulfill an obligation.
Ich habe heute eine Mahnung bekommen, weil ich eine Rechnung vergessen habe.
I received a payment reminder today because I forgot an invoice.
In the meaning 'to urgently call on someone to do something', 'mahnen' (to warn / urge) is often used with the preposition 'zu' + dative. You urge someone toward a particular behavior.
Der Arzt mahnte den Patienten zu mehr Bewegung.
The doctor urged the patient to get more exercise.
When something recalls an event, the fixed structure 'mahnen an' (to serve as a reminder of / commemorate) + accusative is used. The subject here is often a memorial, a place, or a symbol.
Die Ruinen der alten Burg mahnen an eine zerstörerische Schlacht.
The ruins of the old castle serve as a reminder of a destructive battle.
to remember / remind
Erinnern neutrally refers to a possibly forgotten obligation. Mahnen is more forceful and is used especially for payments and debts that are due or already overdue. Before the payment deadline, you can send a friendly reminder; after a default, you can formally mahnen the debtor.

Die Firma mahnt den Kunden schriftlich, weil die Rechnung seit zwei Wochen überfällig ist.
The company sends the customer a written payment demand because the invoice is two weeks overdue.
Die Firma erinnert den Kunden freundlich an die Rechnung, die nächste Woche fällig wird.
The company kindly reminds the customer about the invoice, which is due next week.