Landlords are responsible for the prevention of insect and rodent infestation, and when infestation has taken place, shall be responsible for the extermination of any insects, rodents, or other pests when such extermination is not specifically made the responsibility of the occupant by law or ruling. (see § 309 of the 2009 Uniform Housing Code, adopted by § 21.05.360 of the Lincoln Housing Code).
For example, if a Housing Code investigator determines that a building is infested by cockroaches, s/he may require the landlord to follow a pre-approved extermination program. If the investigator finds that one tenant in the building is causing the pest problem, s/he may cite the tenant and require the tenant to remove garbage or exterminate.
A pest problem should be handled like any other repair problem. The landlord must be informed in writing and you may want to consider involving the Housing Code Office. (See the sections on Repairs.)