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References to 'upper bound on $\vartheta(n)$'
1. $\displaystyle \sum_{n \le x} (\tau(n))^a=O_a(x(\log x)^{2^a-1})$ for $a \ge 0$ by Wkbj79
2. Bertrand's conjecture, proof of by CWoo
3. Wirsing condition by Wkbj79