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Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products

This paper establishes uniform high-order congruences for reciprocal power sums of odd orders and derives an explicit truncated expansion involving complete exponential Bell polynomials for generalized Lehmer-type products, thereby providing a unified framework for computing and verifying these higher-order congruences.

Original authors: Zhenming Tang, Hao Zhong

Published 2026-07-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Zhenming Tang, Hao Zhong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery involving numbers that are trying to hide from each other. In the world of math, there's a special club called "reciprocal power sums." These are just fancy names for adding up fractions like $1/1^2 + 1/2^2 + 1/3^2$ , but with a twist: you only add the ones where the bottom number doesn't share any factors with a specific big number $n$ .

For a long time, mathematicians knew how these sums behaved when you looked at them through a specific kind of "magic lens" called a modulus (specifically, modulo $n$ ). They knew the pattern for the "square" version (where the power is 2). But what about the "odd" versions, like $1/r^3$ or $1/r^5$ ? That was a blank spot on the map.

The Big Discovery: Cracking the Odd Codes
In this paper, authors Zhenming Tang and Hao Zhong decided to fill in that blank spot. They proved that these odd-power sums follow a very specific, uniform rule, just like the even ones do. They didn't just guess; they built a solid bridge using "Bernoulli polynomials" (think of these as a special set of mathematical building blocks) to show exactly how these sums behave modulo $n$ .

They found that if you pick a number $n$ that doesn't share factors with 6 (so, no 2s or 3s in its makeup) and look at an odd power $m$ (like 3, 5, 7, etc.), the sum of these fractions isn't random. It's locked into a formula involving those special building blocks. They even set strict rules for this to work: the power $m$ can't be too big, and it can't be a number that makes the math "glitch" (specifically, $m$ cannot be 1 more than a multiple of $p-1$ for any prime factor $p$ of $n$ ).

The Second Mystery: The Product Puzzle
The paper also tackles a different kind of puzzle: "Lehmer-type products." Imagine you have a giant machine that multiplies a bunch of numbers together, but the machine has a secret switch (the Möbius function) that sometimes cancels things out or flips the sign.

Mathematicians already knew how this machine behaved modulo $n^3$ (a very high level of precision). But what if you wanted to know the answer modulo $n^4$ , $n^5$ , or even higher? The authors found that the old, simple formulas stop working here. You can't just write down a neat, short equation anymore.

Instead, they discovered that to get these higher-order answers, you need to use something called "Bell polynomials." Think of Bell polynomials as a complex, multi-layered recipe. Instead of a single ingredient, you need a whole list of ingredients (the sums we talked about earlier) mixed together in a specific way.

The authors proved that if you use this Bell polynomial recipe, you can expand the product to any level of precision you want (up to $n^{K+1}$ ). They showed exactly how to write this out, giving a clear, step-by-step method to calculate these results. For example, they showed how to get the answer modulo $n^4$ by mixing the first, second, and third power sums together in a specific pattern.

What They Didn't Do (and What They Rejected)
It's important to note what this paper doesn't do. The authors didn't just say, "Hey, maybe this works." They didn't run simulations or guess. They provided a rigorous mathematical proof.

They also explicitly rejected the idea that there is a "comparably simple closed form" for these higher-order product congruences. In other words, you can't find a short, pretty formula like $x = y + z$ for these high-precision results. The complexity is real, and the only way to handle it is through the structured, recursive method using Bell polynomials they developed. They argue that the goal of modern research isn't just to push the modulus higher and higher blindly, but to build a structured framework that is reliable and computable.

The Bottom Line
The paper proves two main things with absolute certainty:

Odd Power Sums: There is a uniform formula for reciprocal sums of odd orders (like $1/r^3$ ) modulo $n$ , expressed using Bernoulli polynomials, provided the numbers meet specific criteria.
Higher-Order Products: For generalized Lehmer-type products, there is no simple shortcut for high-precision answers. Instead, the answer is a structured expansion using complete exponential Bell polynomials, which allows for exact computation at any desired level of precision.

The authors didn't just suggest this; they derived it, proved it, and showed exactly how to use it. They've handed mathematicians a new, powerful toolkit for solving these number theory riddles, turning a chaotic mess of high-order calculations into a structured, solvable game.

Technical Summary: Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products

Problem Statement
This paper addresses two primary questions in the arithmetic of reciprocal sums and combinatorial products modulo high powers of integers. First, it investigates whether existing quadratic reciprocal sum congruences (specifically those involving $1/r^2$ ) can be extended to reciprocal sums of odd order ( $1/r^m$ for odd $m$ ) and expressed uniformly in terms of Bernoulli polynomials for composite moduli. Second, it seeks to expand generalized Lehmer-type products modulo $n^{K+1}$ , moving beyond the known cubic modulus ( $n^3$ ) results to derive higher-order expansions where coefficients are uniformly represented by Bell polynomials.

Methodology
The authors employ a structured approach combining local analysis with global reconstruction:

Local Congruences: The study begins by establishing congruences modulo prime powers $p^l$ . The authors refine existing results by Cai et al. regarding sums of the form $\sum (p^l - tr)^M$ . By utilizing Euler's theorem, Faulhaber's formula, and properties of Bernoulli polynomials, they derive explicit congruences for sums of powers modulo $p^l$ and $p^{2l}$ , distinguishing between cases where the exponent $M$ is even or odd.
Lifting and Inversion: To extend these results to composite moduli, the paper utilizes a "lifting relation" for truncated reciprocal sums. This involves proving that sums over $\lfloor c p^l / e \rfloor$ can be related to sums over $\lfloor p^l / e \rfloor$ via Jacobi symbols. The authors also employ Möbius inversion and the Chinese Remainder Theorem to transition from prime-power results to general composite $n$ .
Bell Polynomial Expansion: For the product congruences, the authors analyze the logarithm of the product in the truncated ring $\mathbb{Z}/n^{K+1}\mathbb{Z}$ . By expanding the logarithm into a power series involving reciprocal power sums $S_m(n)$ and then exponentiating, they identify the resulting coefficients as complete exponential Bell polynomials. This allows the product to be expressed as a finite sum involving these polynomials.

Key Contributions and Results

Extension to Odd-Order Reciprocal Sums (Theorem 1.3):
The paper generalizes previous quadratic results to reciprocal sums of odd order. For a positive integer $n$ with $(n, 6)=1$ , $e \in \{2, 3, 4, 6\}$ , and an odd integer $m$ satisfying specific bounds relative to the prime factors of $n$ , the authors establish the congruence:
$\sum_{\substack{r=1 \\ (r,n)=1}}^{\lfloor n/e \rfloor} \frac{1}{r^m} \equiv n^{\phi(n)-m} \phi_{m-\phi(n)}(n) \left( \frac{B_{\phi(n)-m+1}(1/e)}{\phi(n)-m+1} - \frac{B_{\phi(n)-m+1}}{\phi(n)-m+1} \right) \pmod n$
Here, $B_k(x)$ denotes the $k$ -th Bernoulli polynomial, and $\phi_f(n)$ is a generalized Euler totient function. This result unifies the computation of these sums using Bernoulli polynomials.
Higher-Order Product Expansions (Theorem 1.4):
The paper provides a higher-order expansion for generalized Lehmer-type products. For positive integers $k, e, n, K$ with $(n, K!) = 1$ , the product
$\prod_{d|n} \binom{kd-1}{\lfloor d/e \rfloor}^{\mu(n/d)}$
is shown to satisfy:
$\equiv (-1)^{\phi_e(n)} \sum_{m=0}^{K} \frac{\bar{B}_m(-kS_1(n), \dots, -(m-1)!k^m S_m(n))}{m!} n^m \pmod{n^{K+1}}$
where $\bar{B}_m$ represents the $m$ -th complete exponential Bell polynomial and $S_m(n)$ are the reciprocal power sums.
Explicit Low-Order Cases:
The authors demonstrate the computability of these results by deriving explicit formulas for $K=2$ (recovering previous $n^3$ results) and $K=3$ (providing a new $n^4$ expansion). They also present a recursive algorithm (Equation 53) for computing the coefficients $C_t$ of the expansion, facilitating algorithmic verification.

Significance
The paper claims to provide a unified framework for the explicit computation and algorithmic verification of higher-order congruences. Rather than merely increasing the modulus order for specific cases, the work constructs a structured proof framework that is:

Transferable: Applicable to both reciprocal sums and product expressions.
Recursive: Utilizing Bell polynomials to generate higher-order terms systematically.
Computationally Effective: Offering explicit formulas and recursive relations that allow for the calculation of congruences modulo $n^{K+1}$ without ad-hoc derivations for each order.

The results bridge the gap between classical congruences (like Morley's) and modern structural approaches involving Bernoulli and Bell structures, suggesting that the central task in this field is shifting from isolated refinements to the development of such systematic frameworks.

Technical Summary: Higher-Order Congruence for Reciprocal Power Sums and Generalized Lehmer-Type Products

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