fälschlich
falsely
Based on a false assumption or false information; mistakenly or unjustly.
Er wurde fälschlich des Diebstahls beschuldigt.
He was falsely accused of theft.

falsely
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falsely
Based on a false assumption or false information; mistakenly or unjustly.
Er wurde fälschlich des Diebstahls beschuldigt.
He was falsely accused of theft.

The adverb 'fälschlich' (falsely/mistakenly) is derived directly from the adjective 'falsch' (false/wrong). It belongs to the word family of 'fälschen' (to forge) and 'die Fälschung' (forgery).
Diese Information war von Anfang an falsch.
This information was wrong from the start.
As an adverb, 'fälschlich' (falsely/mistakenly) usually stands directly before the verb or participle it modifies. It emphasizes that the action was based on an error.
Viele Leute haben fälschlich angenommen, das Museum sei heute geschlossen.
Many people mistakenly assumed that the museum was closed today.
'Fälschlich' (falsely/mistakenly) is very often combined with verbs of saying, thinking, or judging. Typical examples are 'fälschlich annehmen' (to mistakenly assume), 'fälschlich behaupten' (to falsely claim), and 'fälschlich beschuldigen' (to falsely accuse).
Die Zeitung behauptete fälschlich, der Minister würde zurücktreten.
The newspaper falsely claimed that the minister would resign.
This adverb belongs more to written language and formal contexts. It is often found in news reports, official reports, or legal texts.
Dem Unternehmen wurde fälschlich vorgeworfen, Steuern hinterzogen zu haben.
The company was falsely accused of having evaded taxes.
correct / right
'richtig' and 'fälschlich' are opposites when it comes to truth or error. 'richtig' says that something is correct or contains no mistake. 'fälschlich', on the other hand, describes something that happens or is said based on a false assumption. 'richtig' can be an adjective, for example with an answer. 'fälschlich' is an adverb and fits especially with actions such as accusing, assuming, or reporting.

Der Kunde wurde fälschlich als säumig gemeldet.
The customer was falsely reported as being in arrears.
Die Antwort der Schülerin ist richtig.
The student's answer is correct.