The Gauss sphere problem concerns the error in counting integer lattice points in a three-dimensional ball.
If 
counts the triples 
with 
,
the main term is the volume 
. The problem is to bound the difference between the
count and this main term.
Bellotti and Kerr (2026) proved that, for every 
,
 |
Here 
denotes big-O notation. This improves the earlier
exponent 
of Heath-Brown (Bellotti and Kerr 2026), since 
.
Bellotti and Kerr (2026) also report the smaller exponent 
using OpenAI's announced 
-quasi-generalized
Riemann hypothesis result, a zero-free half-plane 
for Dirichlet L-series.
That estimate depends on the announced result and is separate from their unconditional
exponent 
.
