Can you talk a bit by exactly what you mean by "customer" here. For example:
- In XP for example "Customer" is a role that's pretty much synonymous with Product Owner in Scrum
- Customer is sometimes used as a synonym for the end-user of the product. So if I'm building an ecommerce system the people buying products would be the customers.
- Customer is sometimes used as a synonym for "client" - so the person who asked you to build the ecommerce system would be the customer.
- Ditto for stakeholder. Ditto for the person paying for the system.
So - which customer will affect the answer considerably ;-)
Customer isn't a Scrum role. It's not a Scrum concept. The PO is the person who owns the responsibility for the product backlog. To quote from the Scrum Guide:
The Product Owner is the sole person responsible for managing the Product Backlog. [...] The Product Owner is one person, not a committee. The Product Owner may represent the desires of a committee in the Product Backlog, but those wanting to change a backlog item’s priority must convince the Product Owner.