The list shall be entered in a well-bound book or loose-leaf book firmly bound, prepared for that purpose, and shall contain appropriate columns in which payments may be credited and the lien of the assessment satisfied by the tax collector of the county. The book shall be known as the assessment book for county improvements and shall be a public record. No error or mistake in regard to the name of the owner shall be held to invalidate any assessment, and it shall be sufficient if the name of the person in whose name the property was last assessed for taxes for state taxation is shown in the book.