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تسجيل مستخدم جديدDistribution mod $p$ of Eulers totient and the sum of proper divisors
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We consider the distribution in residue classes modulo primes $p$ of Eulers totient function $phi(n)$ and the sum-of-proper-divisors function $s(n):=sigma(n)-n$. We prove that the values $phi(n)$, for $nle x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 le p le (log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $ple (log{x})^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$.
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