Well, it only took 34 posts for the OP to tell us, as Paul Harvey used to say "the rest of the story."
If you would have given us that specific info in the beginning, you would have gotten more specific answers based on your application.
Obviously tires used on a closed course do not have to worry about potholes, dips, poor road surfaces, debris etc... And you are not concerned with comfort or tire wear but performance.
You still must know your axle weight and the chosen tires load capacity. The traction characteristics of the chosen tire, speed rating/ load index.
Start with that as a baseline.
You would be using higher cold inflation pressures than for typical passenger cars and proper rim width and suspension mods would also effect preferred air pressure as the tire is a part of the suspension system and things such as wheel/tire weight, spring rates, shock absorber types etc.. all have a combined effect on preferred tire pressures.
Nitrogen is often used because of its larger molecule size which translates to less pressure loss through the liner and that it also displaces the moisture inside the tire retarding the corrosion/oxidation common on alloy wheels and other benefits. Less pressure variation caused by the heat generated in the tire due to flexing when rolling. Higher pressure also help to reduce the slip angles of the tires when cornering helping to maintain stability.
We could go on but the point is, Safety first.
If you are going racing of any type. It is a lot more complex than you think.
This might give you a little more appreciation for the racing teams crew chiefs and technicians. They must have a working knowledge of physics, metallurgy, and many other
disciplines.
Street cars are not race cars . Street cars are designed as a compromise between performance, longevity and comfort. Race cars have short component life, constant maintenance. Something that is unacceptable to the typical motorist.
As mention in the following text, the rim width is also very important. Rule of thumb isn't always accurate. Modern tires have different designed rim widths based on whether the tires are designed with low rolling resistance/increased fuel economy in mind or just the typical dimension formulas used for years.
On radial tires the tread width is typically narrower than the section width of the tire. The old wide tread bias ply tires had a flatter, wider tread contact patch to increase traction.
Radial tires because of the sidewall and tread area working more independently from each other maintains tread contact in corners much better than the old bias ply and needs less tread area to maintain traction . Wheel/Rim Size Calculator - Shows acceptable rim width range for the tyre size