TaoFivePrimes.rosser_schoenfeld_totient_lemma15_small_range

Let $n > 1$ be an integer, $y$ a real number, $\varphi$ Euler's totient function, $\gamma$ the Euler–Mascheroni constant, and $\theta$ the Chebyshev theta function. Assume

Then

This is the small-range companion of Lemma 15 of Rosser and Schoenfeld (1962), §9 (p. 88): the restriction of the platform theorem TaoFivePrimes.rosser_schoenfeld_totient_lemma15 to the range $\log n + y - 2 < 286$, i.e. the finite part of inequality (3.41) below the analytic threshold $x = 286$ of the Mertens-type product bound (3.29). In this range the product route of Theorem 33 followed by (3.29) is not available ((3.29) is only valid for $x \ge 286$), but the inequality remains true: for $\log n + y < 5$ only finitely many $n \le 20$ occur, and in the intermediate range $5 \le \log n + y - 2 < 286$ the extremal values of $n/\varphi(n)$ are attained at primorials $p_k^\#$, and the hypothesis $\log n < \theta(\log n + y)$ together with the cap $y - 2 \le 0.9 \log n / \log\log n$ excludes exactly those near-primorial configurations for which the primorial product $\prod_{p \le p_k} p/(p-1)$ would exceed $e^{\gamma} \log\log n + 5/(2\log\log n)$; for example $n = 23^\#$ is excluded because reaching the next prime $29$ would require $y \ge 29 - \log(23^\#) \approx 9.78$, violating the cap $y \le 2 + 0.9\log(23^\#)/\log\log(23^\#) \approx 7.85$.