The average number of representations of an integer as a sum of two prime powers over multiples of a fixed integer

Abstract: We extend a result by Ikeda and Suriajaya (2025) to find the asymptotic behaviour of the average number of representations of an integer $n$, over multiples of a fixed $q\ge 2$, as a sum of two prime $k$-th powers, for $k\ge 2$.

• The paper establishes a precise asymptotic formula for counting representations of multiples of a fixed integer as sums of two prime k-th powers under GRH.
• It employs Fourier analysis and Dirichlet character expansions to disentangle the main term from error terms, with an explicit factor Σₖ(q) capturing local arithmetic effects.
• A notable finding is that Σₖ(q) may vanish for certain (q, k), highlighting arithmetic obstructions and differences from the classical Goldbach (k=1) scenario.

Asymptotics for Sums of Two Prime Powers over Multiples of a Fixed Integer

Introduction and Context

This paper develops a generalization of prior results concerning the average number of representations of an integer as a sum of two prime powers, specifically over integers which are multiples of a fixed integer $q \geq 2$, and for exponents $k\geq 2$. The foundational case $k=1$ (the Goldbach-type setting) was addressed by Ikeda and Suriajaya, who, under GRH, proved that for $2\leq q \leq N$,

$$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$

where $G_{q,1}(N)$ denotes the average number of Goldbach representations for $n \leq N$, restricted to $n$ divisible by $q$. This paper extends that analysis to general $k$ (power-sum variants) and addresses both the analytic difficulties and arithmetic obstructions introduced as $k\geq 2$0 increases.

Main Result

For any fixed integer $k\geq 2$1 and exponent $k\geq 2$2, under the assumption of the Generalized Riemann Hypothesis (GRH) for Dirichlet $k\geq 2$3-functions modulo $k\geq 2$4, the authors establish the following asymptotic for

$k\geq 2$5

where $k\geq 2$6 counts ordered representations $k\geq 2$7 with each $k\geq 2$8 a prime power (weighted by the von Mangoldt function).

The main result is:

$k\geq 2$9

where $k=1$0 is the unrestricted sum, and $k=1$1 is a multiplicative factor reflecting arithmetic obstructions (described explicitly using Dirichlet characters and congruence conditions).

A notable and nontrivial claim is that $k=1$2 may vanish for certain $k=1$3, reflecting zero average for the counting function on such residue classes, in contrast to the $k=1$4 case where the main term is always present.

Technical Framework and Proof Sketch

The methodology synthesizes Fourier-analytic and character-theoretic techniques:

• The key analytic object is the generating series $k=1$5, which is related to exponential sums over $k=1$6-th powers and primes.
• By Dirichlet character expansion, $k=1$7 is decomposed as an average over characters modulo $k=1$8, with the main term arising only from those $k=1$9 satisfying $2\leq q \leq N$0.
• The use of $2\leq q \leq N$1 introduces strong arithmetic structure: unlike the $2\leq q \leq N$2 case where this sum is always nonzero, for $2\leq q \leq N$3 the possible vanishing arises from the behavior of $2\leq q \leq N$4-th powers modulo $2\leq q \leq N$5.
• The unconditional error analysis depends on GRH, enabling nontrivial bounds on sums involving nonprincipal characters.
• A multiplicativity argument and detailed local analysis provide completely explicit formulae for $2\leq q \leq N$6, showing, for instance, vanishing for certain $2\leq q \leq N$7 and $2\leq q \leq N$8.

The extension from $2\leq q \leq N$9 exploits and generalizes methods from Goldston, Vaughan, Languasco, and Zaccagnini, involving precise analysis of exponential sums and explicit application of Gallagher’s lemma to control mean values.

Arithmetic Obstructions and the Constant $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$0

A central innovation is the explicit computation and interpretation of $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$1 for general $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$2:

• For odd primes $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$3, $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$4 reflects the parity of the index $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$5, and may be zero depending on $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$6.
• For $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$7, more delicate behavior arises, reflecting the structure of the multiplicative group modulo $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$8 and the divisibility properties of $$G_{q,1}(N) = \frac{G_{1,1}(N)}{\varphi(q)} + \mathcal{O}(N \log^3 N)$$9.

Thus, the main term can vanish entirely for dense families of $G_{q,1}(N)$0 for a fixed $G_{q,1}(N)$1, in stark contrast to the $G_{q,1}(N)$2 situation.

Implications and Future Directions

Practical Implications: The formula quantifies the averaged local solubility (over residue classes) of representing an integer by sums of two prime $G_{q,1}(N)$3-th powers. For $G_{q,1}(N)$4 this average may be zero, signaling strong local obstructions and the arithmetic rarity of such representations in certain arithmetic progressions. For analytic number theory, these results determine the uniformity, or lack thereof, of such power-sum representations in arithmetic progressions.

Theoretical Implications: The explicit dependence on local arithmetic, as expressed through $G_{q,1}(N)$5, informs further investigations into higher prime power analogues of Goldbach-type problems, especially in exploring the qualitative differences between the linear ($G_{q,1}(N)$6) and nonlinear ($G_{q,1}(N)$7) regimes. The approach robustly decomposes main and secondary contributions by character theory and orthogonality, which is essential for extensions to more summands or varying coefficients.

Speculation on Future Developments: The methods demonstrated here suggest a route to generalizations involving sums of $G_{q,1}(N)$8 prime $G_{q,1}(N)$9-th powers over multiples of $n \leq N$0, as mentioned in the paper and in ongoing work ([see in-progress work referenced]). There is potential for further refinement in understanding the distribution of representation counts at finer scales, as well as for precise secondary term analysis under even stronger hypotheses. The arithmetic dichotomy for $n \leq N$1 revealed by vanishing $n \leq N$2 may also motivate new investigations into local-global principles for additive problems involving prime powers.

Conclusion

This paper establishes a precise asymptotic for the average number of representations of integers (restricted to multiples of $n \leq N$3) as sums of two prime $n \leq N$4-th powers, for arbitrary $n \leq N$5. The result exhibits significant differences from the $n \leq N$6 case due to new local obstructions, explicitly quantified by the factor $n \leq N$7. The methods and explicit formulae provided form a foundation for further studies in both additive prime number theory and the analytic theory of $n \leq N$8-functions.