We give all functions ƒ , E: ℕ → ℂ which satisfy the relation
for every a, b, c ∈ ℕ, where h ≥ 0 is an integers and K is a complex number. If n cannot be written as a2 + b2 + c2 + h for suitable a, b, c ∈ ℕ, then ƒ (n) is not determined. This is more complicated if we assume that ƒ and E are multiplicative functions.
For a fixed positive integer \kappa, the functional equation \kappa f(m^2 + n^2) = f(\kappa m^2) + \kappa f(n^2), (m,n\in\mathbb{N}) is solved for multiplicative functions f. This complements a 1996 result of Chung [2] which deals with the case \kappa=1. The method used relies on the sum of two squares theorem in number theory.
We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.
Abstract
Let π be an automorphic irreducible cuspidal representation of
GL
m
{\operatorname{GL}_{m}}
over
ℚ
{\mathbb{Q}}
with unitary central character, and let
λ
π
(
n
)
{\lambda_{\pi}(n)}
be its n-th Dirichlet series coefficient.
We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded
conductor, twisted by multiplicative functions
λ
π
(
n
)
{\lambda_{\pi}(n)}
and
μ
(
n
)
λ
π
(
n
)
{\mu(n)\lambda_{\pi}(n)}
. We are able to establish non-trivial bounds for these
algebraic twisted sums with intervals of length of at
least
q
1
/
2
+
ε
{q^{1/2+\varepsilon}}
for an arbitrary fixed
ε
>
0
{\varepsilon>0}
.
Arithmetical Functions Commutable with Sums of Squares II
