The divisor function σₖ(n) sums the k-th powers of all divisors of n. Its 2-adic valuation — how many times 2 divides σₖ(n) — is bounded above by a function of log n. The bound is logarithmic: as n grows, the maximum power of 2 dividing σₖ(n) grows at most logarithmically.
The bound is achieved by Mersenne primes (arXiv:2603.11979). For odd k, the 2-adic valuation reaches its maximum precisely when n is a product of distinct Mersenne primes — primes of the form 2ᵖ - 1. For even k, the maximum is achieved only at n = 3. The arithmetic function's extremal behavior is controlled by a specific class of primes.
The connection to Mersenne primes is not coincidental. When d divides a Mersenne prime M = 2ᵖ - 1, the divisors of M are 1 and M itself. The sum σₖ(M) = 1 + Mᵏ = 1 + (2ᵖ - 1)ᵏ. For odd k, this sum is divisible by 2ᵖ because (2ᵖ - 1)ᵏ ≡ (-1)ᵏ = -1 mod 2ᵖ, making the sum ≡ 0 mod 2ᵖ. The Mersenne structure maximizes the 2-adic valuation because the prime itself is engineered (by being one less than a power of 2) to produce maximal cancellation in the divisor sum.
Products of distinct Mersenne primes inherit this property multiplicatively. Each Mersenne factor contributes its own 2-adic content independently. The extremal values of the divisor function's 2-divisibility are built from the same primes that sit at the boundary between powers of 2 and primality.