ON CONGRUENCES OF THE FORM σ(n) ≡ a (modn)

Abstract

We study the distribution of solutions n to the congruence σ(n) ≡ a ( mod n). After excluding obvious families of solutions, we show that the number of these n ≤ x is at most x½+o(1), as x → ∞, uniformly for integers a with ∣a∣ ≤ x¼. As a concrete example, the number of composite solutions n ≤ x to the congruence σ(n) ≡ 1 ( mod n) is at most x½+o(1). These results are analogues of theorems established for the Euler ϕ-function by the third-named author.

Some problems of Erdős on the sum-of-divisors function

“…The proof uses recent results from [2] on the number of solutions n to congruences of the form σ(n) ≡ a (mod n).…”

“…The main new ingredient is a lemma that is perhaps of independent interest: Almost all primitive nondeficient numbers n have P (n) n 1/3 . We will deduce the lemma from the results of [2] concerning solutions n to the congruence σ(n) ≡ a (mod n). In that paper, n is called a regular solution if (It is straightforward to check that these numbers n are solutions of the congruence.)…”

“…While not phrased this way in [2], the proof of [2, Theorem 1] when a = 0 is handled there by reduction to the convenient case. Indeed, Lemma 5 in [2] asserts that if n is a sporadic solution to σ(n) ≡ a (mod n) and n > 6a 2 log(6|a|), then n is convenient.…”

“…Indeed, Lemma 5 in [2] asserts that if n is a sporadic solution to σ(n) ≡ a (mod n) and n > 6a 2 log(6|a|), then n is convenient. In our setup here, we start by assuming n is convenient.…”

“…Thus, the conclusion of [2, Lemma 5] holds trivially. Moreover, only the convenience hypothesis is needed in the proofs of Lemmas 6 and 7 from [2]. That is, both of those lemmas hold with the assumption of convenience replacing the hypothesis "n is a sporadic solution satisfying n > 6a 2 log(6|a|)".…”