Chebyshev theta function
Chebyshev's theta function, denoted or sometimes , is a function of use in analytic number theory. It is defined thus, for real : where the sum ranges over all primes less than .
Estimates of the function
The function is asymptotically equivalent to ( is the prime counting function) and . This result is the Prime Number Theorem, and all known proofs are rather involved.
However, we can obtain a simpler bound on .
Theorem (Chebyshev). If , then .
Proof. We induct on . For our base cases, we note that for , we have .
Now suppose that . Let . Then so by the inductive hypothesis. Therefore as desired.