Chebyshev theta function

Chebyshev's theta function, denoted $\vartheta$ or sometimes $\theta$ , is a function of use in analytic number theory. It is defined thus, for real $x$ : $\vartheta(x) = \sum_{p \le x} \log p ,$ where the sum ranges over all primes less than $x$ .

Estimates of the function

The function $\vartheta(x)$ is asymptotically equivalent to $\pi(x)$ (the prime counting function) and $x$ . This result is the Prime Number Theorem, and all known proofs are rather involved.

However, we can obtain a simpler bound on $\vartheta(x)$ .

Theorem (Chebyshev). If $x \ge 0$ , then $\vartheta(x) \le 2x \log 2$ .

Proof. We induct on $\lfloor x \rfloor$ . For our base cases, we note that for $0 \le x < 2$ , we have $\vartheta(x) = 0 \le 2x \log 2$ .

Now suppose that $x \ge 2$ . Let $n = \lfloor x \rfloor$ . Then $2^x \ge 2^n \ge \binom{n}{\lfloor n/2 \rfloor} \ge \prod_{\lfloor n/2 \rfloor < p \le n} p ,$ so $x \log 2 \ge \sum_{\lfloor n/2 \rfloor < p \le n} \log p = \vartheta(x) - \vartheta(\lfloor n/2 \rfloor) \ge \vartheta(x) - 2\lfloor n/2 \rfloor \log 2 \ge \vartheta(x) - x \log 2 ,$ by inductive hypothesis. Therefore $2x \log 2 \ge \vartheta(x),$ as desired. $\blacksquare$

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Chebyshev theta function

Estimates of the function

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