G-Additive Functions Overview

• G-additive functions are arithmetic and group functions that satisfy additive properties—such as f(mn)=f(m)+f(n) for coprime m and n—and include cases with uniform prime assignments.
• They are pivotal in analytic number theory, with techniques like Vinogradov’s estimate and Vaughan’s identity used to derive tight bounds for associated exponential sums.
• In group theory, these functions illustrate decomposability over abelian groups, offering insights into arity gaps and optimal conditions for additive bases.

A G-additive function is a concept arising in several distinct but related contexts within arithmetic and group theory, referring either to special classes of additive arithmetic functions or to function decomposability properties on abelian groups. Prominent manifestations include the class $F_0$ of additive arithmetic functions with uniform prime values, functions defined via polynomial assignments at primes, and decomposability into lower-arity summands respecting abelian group structure.

1. Classes and Definitions of G-Additive Functions

Multiple frameworks for "G-additive" functions are in current usage:

• Arithmetic G-additive functions ($F_0$): An arithmetic function $f:\mathbb N\to\mathbb C$ is in the class $F_0$ if it is additive (i.e., $f(mn) = f(m) + f(n)$ for $(m, n) = 1$) and satisfies $f(p) = 1$ for every prime $p$ (Gafni et al., 7 Feb 2025).
• Polynomially-defined or G-additive functions: Any arithmetic function $g:\mathbb N \to \mathbb Z$ with $g(mn) = g(m) + g(n)$ for $(m, n) = 1$ and such that there exists a nonconstant polynomial $G(T)$ with $g(p) = G(p)$ for every prime $p$ (Roy, 2023).
• Group-theoretic G-additive functions: For an abelian group $G$, a function $f:G^n \to G$ is termed G-additive (or additively decomposable over $G$) if it admits a decomposition as a sum of functions, each depending on fewer than $n$ variables (Couceiro et al., 2011).
• G-additive functions in additive bases: The notation “G-additive functions” also appears in the context of extremal functions (e.g., $E_G, X_G, S_G$) related to additive bases in groups (Lambert et al., 2015).

A summary of key classes:

Context	Condition	Examples
Arithmetic, $F_0$	Additive, $f(p) = 1$ for all primes	$\omega(n)$, $\Omega(n)$
Polynomially-defined	Additive, $f(p) = G(p)$ for some nonconstant $G$	$\Omega(n)$ ($G=1$), $\sum_{p|n}p$
Abelian group functions	Decomposable into lower-arity functions over $G$	Oddsupp functions, arity-gap 2 cases

2. Exponential Sums and Analytic Techniques for $F_0$ Functions

A core analytic development is the establishment of tight bounds for exponential sums of the form

$$S_f(\alpha; X) = \sum_{n \leq X} f(n) e(\alpha n), \quad f \in F_0$$

where $e(t) = \exp(2\pi i t)$. The primary result states that, for any real $\alpha = a/q + \theta$ with $(a, q) = 1$ and $|\theta| \leq q^{-2}$, and any $\Delta \in (0, \frac12)$,

$$S_f(\alpha; X) \ll \frac{X}{q^\Delta + X^{5/6} + X^{1-\Delta}q^\Delta} \left((\log X)^4 + (\log X) F_f(X)\right)$$

where $F_f(X) = \max_{p^\ell \leq X} |f(p^\ell)|$. The proof strategically splits the sum into contributions from small and large prime powers, harnessing classical tools such as Vinogradov’s estimate and Vaughan’s identity, and optimizing over a transition parameter at $X^{5/6}$ (Gafni et al., 7 Feb 2025).

Concrete examples within $F_0$ are:

• $f(n) = \omega(n)$: $F_f(X) = 1$, yielding $(\log X)^4$ as the principal secondary factor.
• $f(n) = \Omega(n)$: $F_f(X) = \lfloor \log X \rfloor$, incurring one additional logarithmic factor in the bound.

3. Distribution Results for Polynomially-defined Additive Functions

For a vector of polynomially-defined additive functions $g_1, ..., g_M$ defined by nonconstant polynomials $G_1, ..., G_M$, a uniform joint equidistribution theorem is established. Provided the collection $\{\widetilde G_j = G_j - G_j(0)\}$ is $\mathbb Q$-linearly independent, for jointly varying moduli $q \leq (\log x)^K$ and every residue class tuple $\mathbf{a} = (a_1, ..., a_M)$ modulo $q$,

$$N(x; q, \mathbf{a}) = |\{n \leq x: g_j(n) \equiv a_j \pmod{q} \ \forall j\}| = \frac{x}{q^M}(1 + O((\log x)^{-B}))$$

for suitable error exponent $B < K$ (Roy, 2023). The proof utilizes prime factorization sieves, additive character orthogonality, and uniform exponential sum estimates (Weil, Cochrane–Zheng). This extends Delange's fixed-modulus criterion to a wide range of moduli and to the joint setting.

4. Additive Decomposability and Arity in Group-Valued Functions

In the context of functions $f:G^n \to G$ on abelian groups, G-additivity refers to decomposability: $f$ is G-additive if it can be written as a sum of functions each depending on fewer than $n$ variables. A central result is the classification by arity gaps:

• For a finite abelian group $G$, all oddsupp-determined functions (those with arity gap 2) are decomposable if and only if the exponent of $G$ is a power of 2 (Couceiro et al., 2011).
• The derivative criterion provides an explicit check for decomposability: all full-order partial derivatives at the origin must vanish for $(n-1)$-decomposability.

In particular, oddsupp-based functions (determined by the multiset of variables occurring an odd number of times) serve as extremal examples both for possible and impossible decompositions, depending on the exponent structure.

5. Extremal G-Additive Functions in Additive Bases of Groups

When studying additive bases in infinite abelian groups, several extremal G-additive functions are defined:

• $E_G(h)$: maximal number of exceptional elements (whose removal destroys the basis property) in a basis of order $h$, with $E_G(h) \leq h-1$ always.
• $X_G(h)$: maximal order required for $A \setminus \{a\}$ when removing a regular element $a$ from a basis $A$ of order $h$; bounds range from linear in pure torsion groups to quadratic in groups with $\mathbb Z$ quotients.
• $S_G(h)$: minimal $s$ such that all but finitely many elements of every basis of order $h$ can be removed without the order exceeding $s$, with $h+1 \leq S_G(h) \leq 2h$ for any infinite abelian $G$ (Lambert et al., 2015).

These functions expose the interplay between additive structures and the ambient group's algebraic properties, with explicit behaviors depending on torsion, divisibility, and the presence of free parts.

6. Applications: Goldbach-Type Problems and Short-Interval Behavior

The minor-arc control of exponential sums involving G-additive functions yields concrete analytic number theory results. Notably, for $$r_\Omega(N) = \sum_{n_1 + n_2 + n_3 = N} \Omega(n_1)\Omega(n_2)\Omega(n_3)$$, the circle method and the bound for $S_f(\alpha; X)$ establish

$$r_\Omega(N) = \frac{N^2}{2} \mathfrak S(N, M) + O\left(N^2 (\log\log N)^3/(\log N)^A\right)$$

where $\mathfrak S(N, M)$ is an explicit singular series built from Ramanujan sums and Dirichlet convolutions (Gafni et al., 7 Feb 2025).

In "short interval" contexts, the average behavior and gap statistics for additive functions are tightly linked: any nontrivial bound on average gaps between $g(n)$ and $g(n-1)$ implies (and is implied by) corresponding bounds for centered moments in the values of $g$ (Mangerel, 2021). These results have ramifications for rigidity phenomena (e.g., Erdős’s almost-everywhere monotonicity conjecture for additive functions).

7. Extensions and Further Directions

The analytic framework for $F_0$ and polynomially-defined additive functions is broadly extensible:

• To wider classes $F_b$ with prime assignments $f(p) = p^b$.
• To exponential sums of the form $\sum_{n \le X} z^{f(n)} e(\alpha n)$, which include twisted divisor-type sums.
• To partition generating functions with exponents indexed by G-additive functions, via the circle method.
• To sharper explicit formulas under the Riemann Hypothesis, associating sums over $z^{f(n)}$ with zero distributions of $\zeta(s)$ (Gafni et al., 7 Feb 2025).

In group-theoretic directions, the full characterization of when all gap-2 (oddsupp) functions are decomposable shows a precise group-theoretic threshold at the exponent, thus connecting decomposition theory with structural group invariants (Couceiro et al., 2011).

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References:

• (Gafni et al., 7 Feb 2025) Exponential sums weighted by additive functions
• (Roy, 2023) Joint distribution in residue classes of families of polynomially-defined additive functions
• (Mangerel, 2021) Additive functions in short intervals, gaps and a conjecture of Erdős
• (Couceiro et al., 2011) Additive decomposability of functions over abelian groups
• (Lambert et al., 2015) Additive bases in groups