Let f:N→N be multiplicative. Determine all such functions for which the set of local maxima n≥ 2:f(n)>f(n-1) and f(n)>f(n+1) is finite, and determine all such functions for which the set of local minima n≥ 2:f(n)<f(n-1) and f(n)<f(n+1) is finite.

Sárközy’s local-extrema conjectures

Let $f:\mathbb{N}\to\mathbb{N}$ be multiplicative. Determine all such functions for which the set of local maxima $\{n\ge 2:f(n)>f(n-1)\text{ and }f(n)>f(n+1)\}$ is finite, and determine all such functions for which the set of local minima $\{n\ge 2:f(n)<f(n-1)\text{ and }f(n)<f(n+1)\}$ is finite.

References

Primary source

arXiv

Additional references

On Sárközy's Local Extrema Conjectures — arXiv — Alexander P. Mangerel

analytic-number-theory multiplicative-functions

Posted 14 hours ago · edited 13 hours ago

Progress summary

Refreshed 13 hours ago

Claimed progress

A new preprint reports a broad classification of the relevant patterns and claims to settle both conjectures, but the claimed resolution is not independently verified.

Sárközy’s two conjectures concern local-extrema patterns of multiplicative functions and associated finiteness questions.

Recent classification preprint

Alexander P. Mangerel’s preprint, On Sárközy’s Local Extrema Conjectures, introduces a logarithmic-density classification of local-extrema patterns and applies it to the two finiteness questions. Its claim to settle both conjectures has not received independent mathematical assessment, so the work establishes a claimed advance rather than a verified resolution.

Current status (as of October 2026): The conjectures remain open in the verified literature; Mangerel’s preprint claims a complete resolution, but that claim is unverified.

Sources

No solutions have been posted yet.