Sorry for the tardy reply.
You live in Chicago. But, we need to know:
1. How many units in your building?
2. Does the landlord live in your building?
If your building or complex consists of seven units or more, then you are covered by the Chicago Residential Landlord and Tenant Ordinance ("RLTO"). And, if your buidling is six units or less AND if the landlord does NOT live in the building, then you are also covered by the RLTO.
Under the RLTO, the landlord has a specific duty and responsibility (if the building is covered) to provide an insect/roach free environment.....and if the landlord does NOT do so, you have remedies, such as repair and deduct or paying reduced rent....but to do so, one must TOTALLY AND STRICTLY comply with the modest requirements of Section 110 of the RLTO. AGAIN, this applies IF the building in whiich you live is either seven units or more, OR, if six units or less as long as the building is NOT owner-occupied.
Further, the City of Chicago Building Code demands that the landlord keep the building in good condition and repair, and complaining to the City of Chicago Building Department would be appropriate. In such a case, even if you contact the City via telephone call, you SHOULD/MUST, that very day, confirm your conversation and complaint via a letter to the City of Chicago Building Department and an indicated copy of that letter should be sent and/or delivered to your landlord ..... and of course, keep copies of the letter. The purpose of notification to your landlord is to CONFIRM and put your landlord on actual notice that you have made such a complaint to the City of Chicago.
As to how to notify the landlord, I suggest that you go to the post office and get a "Certificate of Mailing" and use that method to send the landlord's copy to the landlord.....this is NOT certified mail, return receipt requested - which would be the BEST method of notifying your landlord in writing of the problem....I find that many landlords will just NOT accept or pick up certified mail, return receipt requested. The Certificate of Mailing confirms, via the U.S. Post office, that on a given day you sent a letter to a given person....to be sure, the post office does not know what was in the envelope.
If you want to be double-cautious, send both certified mail RRR, and another copy voa certificate of mailing.
You can bet that when the landlord gets such a formal demand to correct the roach problem and/or (if you are covered by the RLTO), that you intend to reduce your rent if after notice the landlord does not correct the problem) and particularly with a month to month tenancy, that the landlord will probably start an eviction proceeding in court. So, think about your options carefully.
If you ARE covered by the RLTO, get a copy in City Hall...they are free....and read about your rights. One of the very good things about the RLTO is that if the landlord acts badly, or does not do repairs and especially if the landlord RETALIATES -- tries to evict you....after you have made a proper demand for repairs, then if you hire a lawyer and win your counter-claim for retaliatory eviction, the legal fees of the tenant are to be paid by the landlord!
Get to know your rights, you may want to contact a tenant's rights organization like the Metropolitan Tenants Organization in Chicago, and/or consult with an attorney of your choice...in any event, you do have rights and the power to cause positive change.