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It’s not a problem, it’s a feature I would say after my research, because this issue is reported in posts here since 2008…
I am using Magento 1.6.2 and encounter the same problem: PayPal Invoice is created but not sent via email.
How can I achieve that paypal invoices are sent automatically after creation?
You can recheck your magento config. After logging in backend, please go to System->PaypalMethod->Authorize.net and set Email Customer as Yes to have email sent to customer after invoice.
Hope you succeed.