Define \( \theta(x) = \sum_{p \le x} \log p\). To prove Bertrand's postulate it is sufficient to prove that \( \theta(2 x ) - \theta(x) > 0 \) for sufficiently large \( x\) and check smaller values of \(x \) by hand.
Remarks:
- One could calculate the asymptotic expansion of \( \theta(x) \) and then verify that \( \theta(2 x ) - \theta(x) > 0 \). However, I think this is more difficult because knowing the asymptotic expansion is already closely related to the prime number theorem.
- I use the terminology "derivative" of \(f(x) \) for expressions like \( f(x) - f(\frac{x}{2} ) \)
It turns out that it is easier to treat the Möbius transform \( \psi(x) = \sum_{n=1}^\infty \theta(x^{1/n}) \). From bounds on "derivatives" of \( \psi(x) \) Ramanujan deduces bounds on the "derivatives" of \( \theta(x) \) . I call this the Tauber step.
To obtain bounds on "derivatives" of \( \psi (x) \), Ramanujan uses the Möbius transform \( \log \lfloor x \rfloor ! = \sum_{n=1}^\infty \psi(\frac{x}{n} ) \) and uses bounds on "derivatives" of \( \log \lfloor x \rfloor ! \), and another Tauber step. Finally, bounds on \( \log \lfloor x \rfloor ! \) are easy because one has Stirling's approximation.
Hence, the structure of the proof is
Remarks:
To obtain bounds on "derivatives" of \( \psi (x) \), Ramanujan uses the Möbius transform \( \log \lfloor x \rfloor ! = \sum_{n=1}^\infty \psi(\frac{x}{n} ) \) and uses bounds on "derivatives" of \( \log \lfloor x \rfloor ! \), and another Tauber step. Finally, bounds on \( \log \lfloor x \rfloor ! \) are easy because one has Stirling's approximation.
Hence, the structure of the proof is
\(
\begin{array}{ccccc}
\text{bounds on} & & \text{bounds on} & & \text{bounds on}\\
\text{(derivatives of)} & \xrightarrow{T} & \text{(derivatives of)} & \xrightarrow{T}& \text{(derivatives of)}\\
\log \lfloor x \rfloor ! && \psi(x) && \theta(x)
\end{array}
\)
Remarks:
- Here \( \xrightarrow{T} \) means the Tauber step.
- What I call the Tauber step is a simple version of the Landau-Ingham Tauberian theorem, see notes from Michael Müger for details on this theorem.
- In the next post I will follow Ramanujan's manipulations to calculate bounds in a simpler situation.
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