How many primes are there below n? Testing the prime number theorem

AA SLAA HLAI SL Solid Also in: Statistics, Modelling

Research question

How accurately do x/ln x and the logarithmic integral approximate the number of primes below x for x up to 10⁷, and how does the relative error behave?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

Primes look random, yet their count follows a smooth law. Testing the approximations yourself, and measuring how slowly the error shrinks, connects number theory with modelling.

The mathematics you'll need

Counting functions and logarithms
Relative and absolute error
Integration: the logarithmic integral (explain; evaluate numerically)
Modelling error against x on log scales

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Generate primes with a sieve (spreadsheet or program); check counts against OEIS.

OEIS (On-Line Encyclopedia of Integer Sequences) — Check a sequence you have found and read its known formulas and references.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain the sieve and count primes.
Compare with x/ln x.
Compare with Li(x).
Model the relative error.
Reflect on why the prime number theorem is about ratios, not differences.

Pitfalls that cost marks

Code with no explanation.
Confusing log bases.
Presenting tables without interpretation.

Showing personal engagement

Explore twin primes or primes in arithmetic progressions.
Make a prediction for 10⁸ and check with published counts.
Explain the theorem's history briefly.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Investigate the average gap between primes near x and compare with ln x.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).