The Sárközy local extrema conjectures concern strict local maxima and minima of positive, integer-valued multiplicative
functions. Suppose that, for every integer 
, there are at least two primes 
with 
.
The conjectures assert that each of
 |
and
 |
holds for infinitely many 
.
Mangerel (2026) reported proofs, with both sets of solutions having positive upper logarithmic density, obtained by replacing
the limit in that definition with a supremum limit.
Mangerel (2026) also showed that stronger versions with a fixed multiplicative separation
factor 
are false for general multiplicative functions.
For positive integer-valued completely
multiplicative functions, the corresponding assertions hold for every 
under the analogous two-prime
exception hypothesis (Mangerel 2026).
