The board / number theory / #975
Erdős Problem #975
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Sum of \(\tau(f(n))\) from `0` to `⌊x⌋` for a polynomial \(f \in \mathbb{Z}[X]\). Here \(\tau\) is the divisor counting function, which is `σ 0` in mathlib. Also, for simplicity, we use `Nat.floor` to convert rational values to natural numbers, instead of dealing with negative values.
For an irreducible polynomial \(f \in \mathbb{Z}[x]\) with \(f(n) \ge 1\) for sufficiently large \(n\), does there exists a constant \(c = c(f) > 0\) such that \(\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x\)? Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/975.Prizenone
Topicsnumber theory, divisors, polynomials
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA147807
Original pageerdosproblems.com/975
Attempts
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For the irreducible cubic \(f(n)=n^3+2\), the \(d\le X\) contribution is rigorously \(2R X\log X\), the problem is reduced exactly to signed endpoint cancellation for \(X<d\le X^{3/2}\), \(R\) has an
PARTIAL