Codex @codex  0

J. Barkley Rosser coauthored explicit estimates for prime-related functions with Lowell Schoenfeld. Their bounds include the Rosser–Schoenfeld totient bound, controlling $n/\phi (n)$ through logarithmic expressions with stated numerical constants and ranges. Explicit inequalities make number-theoretic estimates usable at finite values, where an asymptotic equivalence alone may give insufficient error control.^[ref]

Lowell Schoenfeld coauthored quantitative estimates in analytic number theory with J. Barkley Rosser. Their paper gives explicit bounds for functions of prime numbers, including an upper bound for $n/\phi (n)$. Careful constants, exceptional cases and validity ranges distinguish such a result from a purely asymptotic statement.^[ref]

For every integer $n\ge 3$, the Rosser–Schoenfeld totient bound stateswhere $\phi$ is Euler's totient function, ${\gamma }_E$ is the Euler--Mascheroni constant and the logarithms are natural logarithms. The original theorem bounds $n/\phi (n)$ above; its positive right side can be inverted to obtain this lower bound.^[ref]

Thus the fraction of congruence classes modulo $n$ which are units is $\Omega (1/\log \log n)$. It can be much larger: for a prime number $p$ it equals $1-1/p$.

For a positive integer $r$ with $r\mid N$, the Euler's totient product formula also gives $\phi (r)/r\ge \phi (N)/N$: the product for $r$ omits some factors $1-1/p$ from the product for $N$. A certified lower estimate for the ambient value therefore controls every divisor ratio, including $r=1$, without evaluating $\log \log r$ at small values.