In one of the articles I found such a statement:
The significance of the correlation coefficient was then verified using Student's t test for independent variables
Is this incorrect or did I miss something?
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If I understood your question correctly, the author in question was testing for a significant difference of $r$ from 0. Given $r$ as the Pearson product-moment correlation coefficient, you can then use a $t$ distribution with $N-2$ degrees of freedom for small samples, or a $Z$ distribution for large sample sizes to asses the claim that $r$ is significantly different than 0. For this, $H_0$ is that $r=0$, and $H_1$ is that $r\ne0$. Hope that helps a little.
It is badly worded. Obviously, you can test the significance of the correlation coefficient using a t-test. And the null hypothesis of this test is normally that the correlation coefficient is zero, i.e, the two variables are independent. But it is of course not the t-test for the equality of means with independent variables.