Gruber P. Convex and Discrete Geometry

410 Geometry of Numbers
Collecting the work of Rankin [823], Cassels [194], Ennola [303], Diananda [266]
and Montgomery [752] (global minima) and Ryshkov [864] (local minima) we state
the following result.
Theorem 27.1. The following propositions hold:
(i) Let s > 1. Then ζ(·, s) attains its global minimum among all lattices in E 2 of
determinant 1 precisely for the regular hexagonal lattices.
(ii) Let s ≥ 3. Then the only local minimum of ζ(·, s) among all lattices in E 2 of
determinant 1 is the global minimum.
The 3-dimensional case was studied by Ennola [304] and Sandakova [876].
27.3 Lattice Points in Large Convex Bodies
A classical problem of analytic number theory, the Gauss circle problem [366], is to
estimate the number of points of the integer lattice Z 2 in the circular disc ρ B 2 for
large ρ>0. Clearly,
A(ρ) = #(Z 2 ∩ ρ B 2 ) = ρ 2 π + O(ρ) as ρ →∞.
O(ρ) may be replaced by O � ρ 2 �
3 asshownbySierpiński [938]. The best upper
estimate, at present, is due to Huxley [533], who showed that instead of O(ρ) one
may put O � ρ 131
208 (log ρ) 18637 �
4160 . It was proved by Hardy [479] that O(ρ) may not be
replaced by o � ρ 1 �
2 . The best pertinent result known was given by Soundararajan [949].
In an article which is not yet published, Cappell and Shaneson [189] seem to have
proved the bound O (ρ 1 2 +ɛ ).
The circle problem was the starting point of a voluminous literature in the context
of analytic number theory, see the survey of Ivić, Krätzel, Kühleitner and Nowak
[540], the books of Fricker [344] and Krätzel [616] and the short report in Gruber
and Lekkerkerker [447].
In this section we present a result of Hlawka [511] which is representative of this
area. Since all available proofs are technically involved and require sophisticated
analytic tools, no proof is given.
Hlawka’s Lattice Point Theorem
Theorem 27.2. Let C be an o-symmetric convex body, the boundary of which is of
class C 2 and has positive Gauss curvature. Then
A(ρ) = #(Z d ∩ ρC) = ρ d V (C) + O � 2
d−2+ �
ρ d+1 as ρ →∞.
Here O � 2
d−2+ � � d−1 �
ρ d+1 cannot be replaced by o ρ 2 .
For various refinements due to Krätzel, Nowak, Müller and others, see the survey of
Ivić, Krätzel, Kühleitner and Nowak [540].