The U.S. State Department estimates there are about 7.6 million American expats living abroad, excluding military personnel, but the Internal Revenue Service typically hears from fewer than a million of them, indicating that many who owe U.S. taxes aren't paying them. The IRS has no idea how many are breaking the law.

"To our knowledge, the IRS does not have the ability to estimate how many expats may have a filing requirement," says Kenneth J. Drexler, senior adviser to the Taxpayer Advocate of the IRS.

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