Mixed character sums modulo prime powers

Abstract: We obtain explicit estimates for the mixed character sum $$S= S(χ,g,f,p^m) = \sum_{x=1}^{{p^m}} χ(g(x)) e_{p<sup>m}(f(x))$$, where $p<sup>m$ is a prime power, $χ$ is a multiplicative character mod $p<sup>m$ and $f,g$ are rational functions over $\mathbb Q$. Let $f=f_+/f_-$, $g=g_+/g_-$ in reduced form, and set $D=\text{deg}(f)+Z-1$ where $Z$ is the number of distinct complex zeros of $f_-g_+g_-$, and $Δ= \text{deg}(f)+\text{deg}(g)$ for polynomial $f,g$, $Δ=2(\text{deg}(f)+\text{deg}(g))$ otherwise. We show for example that for odd $p$, any non-degenerate sum has $$|S|\le 3^{4/3}\, p^{m(1-\frac 1D)}$$ if $\text{deg}_p(f) \ge 1$, and $$|S| \le 3^{4/3}\, p^{m(1-\frac 1Δ)}$$ if $\text{deg}_p(g) \ge 1$. Analogous bounds are given for degenerate sums.

• The paper develops explicit uniform upper bounds for mixed character sums, generalizing classical exponential sum estimates.
• It introduces an algebraic parameter D that captures key invariants of the rational functions, yielding sharper cancellation exponents.
• The study employs p-adic Taylor expansion and critical point analysis, addressing both non-degenerate and degenerate cases comprehensively.

Mixed Character Sums Modulo Prime Powers: An Expert Synthesis

Introduction and Context

The paper "Mixed character sums modulo prime powers" (2604.02614) by Cochrane and Granville develops explicit and uniform upper bounds for mixed character sums of the form

$$S(\chi, g, f, p^m) = \sum_{x=1}^{p^m} \chi(g(x)) e_{p^m}(f(x)),$$

where $p^m$ is a prime power modulus, $\chi$ is a multiplicative character modulo $p^m$, and $f, g$ are rational functions over $\mathbb{Q}$. The work generalizes and sharpens classical bounds for pure exponential sums (i.e., when $g=1$), subsuming results of Mordell, Hua, Weil, and others while extending the analytic framework to mixed character sums involving nontrivial multiplicative twists and rational function arguments. The motivation extends to applications in analytic number theory, particularly in the context of additive problems with congruence constraints related to higher prime power moduli, and control over mixed character sums with precise dependence on the algebraic features of $f$ and $g$.

Main Results: Uniform Bounds via Algebraic Invariants

Non-Degenerate Sums

The core contribution is the derivation of explicit upper bounds for $|S(\chi, g, f, p^m)|$ in terms of algebraically meaningful parameters:

• $p^m$0, where $p^m$1 counts the distinct complex zeros of $p^m$2 (with $p^m$3, $p^m$4 in lowest terms);
• $p^m$5 (for polynomials) given by $p^m$6, and for general rational functions, $p^m$7.

For odd $p^m$8 and any non-degenerate sum (i.e., unless $p^m$9 is constant modulo $\chi$0 and $\chi$1 degenerate w.r.t. $\chi$2), they prove:

$\chi$3

This provides significant improvement over previously known estimates in two key aspects: (1) explicit uniform constants, and (2) the dependence of the exponent on the algebraic invariant $\chi$4 rather than on the total degree or a more coarse parameter. Notably, $\chi$5 can be substantially less than $\chi$6 when there is coalescence of zeros among $\chi$7, $\chi$8, or $\chi$9.

The authors also analyze the sharpness of the exponent. For example, in diagonal cases such as $p^m$0, the estimate reduces to the classical $p^m$1 bound, which is known to be tight.

Degenerate Sums

Degenerate cases, including when $p^m$2 is $p^m$3-adically constant or $p^m$4 is a perfect $p^m$5-th power (with $p^m$6 the order of $p^m$7), may reduce to character sums over lower modulus or even become constant. The analysis yields new uniform bounds with additional factors of $p^m$8, where $p^m$9 quantifies the level of degeneracy within the sum. This extended treatment, absent from earlier work (cf. [Cochrane 2002]), clarifies the behavior of all parameter regimes.

Local Analysis and Critical Points

A refined analysis employs Taylor expansion and critical point theory, wherein the main term contributions to $f, g$0 correspond to solution sets of the "critical point congruence":

$f, g$1

with multiplicity $f, g$2 for each critical point. The local contribution associated to a critical point of multiplicity $f, g$3 is bounded by $f, g$4 up to explicit constants, interpolating between the classical square-root cancellation in the generic case and weaker cancellation at higher-multiplicity points.

Technical Approaches and Innovations

Lifting and Deformation

Key tools involve $f, g$5-adic Taylor expansion, evaluation of the mixed character via $f, g$6-adic logarithms, and the reduction of the original mixed sum to nested sums: first over (lifted) critical points, then over smaller moduli where pure exponential sum theory applies.

Explicit Decompositions

The derivation of the parameter $f, g$7 is rooted in a careful algebraic decomposition of the rational functions and their derivatives, including the identification of zeros and the explicit tracking of conductor and primitivity of the multiplicative character.

$f, g$8 Case

Special treatment is given for $f, g$9, notably in the detailed analysis of the structure of multiplicative characters and careful control of the $\mathbb{Q}$0-adic logarithm, which requires deeper attention compared to odd primes due to the presence of higher ramification.

Implications and Future Outlook

The results have both theoretical and practical implications:

• Theoretical: The uniformity in upper bounds with respect to naturally defined algebraic parameters $\mathbb{Q}$1 and $\mathbb{Q}$2 strengthens the analytic machinery available for questions involving equidistribution, cancellation in character sums, and analysis on higher-dimensional or more singular algebraic sets.
• Practical/Algorithmic: Precise knowledge of cancellation phenomena in $\mathbb{Q}$3 supports applications to the distribution of values of rational functions in arithmetic progressions, the study of arithmetic combinatorics in finite rings, and could have further connections to $\mathbb{Q}$4-adic analysis and arithmetic geometry.

The explicit nature of the estimates, including explicit constants and the case-by-case treatment of degeneracies, makes the bounds readily portable to analytic number theory problems requiring precise error terms. Of note is the applicability of these results to incomplete sums via reduction to the complete case, as in the authors' related work [CGZ, in preparation].

There remain open questions concerning the extension of the sharpest bounds to broader classes of functions—especially rational functions with substantial cancellation among zeros or for higher-degree multiple poles/zeros—and determining analogous (best possible) exponents for more general or constrained classes of sums.

Conclusion

This work provides a comprehensive, explicit framework for bounding mixed character sums modulo prime powers, with advances both in algebraic parametrization and explicitness of constants. By introducing the $\mathbb{Q}$5 parameter as a central invariant and conducting a refined local analysis around critical points, significant improvements and clarifications are achieved over the existing literature. The approach not only resolves open aspects of the theory but also creates a foundation for further generalizations and applications in analytic number theory and related domains.