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Counterexamples of Friedlander--Iwaniec dual sums conjecture

This paper refutes the Friedlander–Iwaniec conjecture on the uniform boundedness of sharply truncated nonlinear dual sums by constructing counterexamples using the $m$ -th power of the Riemann zeta function for $m \geq 4$ .

Original authors: Khai-Hoan Nguyen-Dang

Published 2026-07-21

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Khai-Hoan Nguyen-Dang

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 trying to count the number of ways to build a tower out of blocks. In the world of numbers, this is like asking: "How many different ways can I multiply whole numbers together to get a specific result?" For example, the number 12 can be made by $3 \times 4$ , $2 \times 6$ , or $1 \times 12$ . Mathematicians call these "divisor functions," and they are the building blocks of a vast, mysterious landscape called analytic number theory.

To navigate this landscape, scientists use powerful tools called "Dirichlet series." Think of these as magical telescopes that turn a messy pile of numbers into a smooth, flowing wave. When you look at these waves through the telescope, they often follow a strict rule called a "functional equation," which acts like a mirror: if you flip the wave over, it looks almost the same, just with the numbers swapped. This mirror trick is so useful that it allows mathematicians to predict how the numbers behave far out in the distance, even when they can't count them one by one.

For a long time, a group of brilliant mathematicians, including John Friedlander and Henry Iwaniec, proposed a bold idea about these waves. They suggested that if you look at a specific, jagged slice of this wave (a "dual sum"), the jaggedness should cancel itself out perfectly. They predicted that no matter how far you zoom in or how many numbers you count, the leftover "noise" would be so tiny it would practically vanish. It was a beautiful, elegant guess that promised to simplify some of the hardest puzzles in math.

But in this paper, a mathematician named Khai-Hoan Nguyen-Dang steps forward with a reality check. Using a specific type of number tower (the product of the famous Riemann zeta function with itself, written as $\zeta(s)^m$ ), the author constructs a scenario where that beautiful cancellation simply doesn't happen. Crucially, this failure occurs for towers with four or more layers ( $m \ge 4$ ), but the specific way the failure manifests depends on the height of the tower. For $m=4$ , the counterexample works under a specific range of conditions. However, for taller towers with $m=5$ or $m=6$ , the author proves that the prediction fails even more broadly, including in the standard range of numbers usually studied. Instead of vanishing, the noise turns out to be loud and persistent. The paper proves that for these specific large numbers, the "jaggedness" is actually a massive, predictable wave that refuses to disappear. This isn't just a small error; it's a fundamental crack in the prediction, showing that the elegant rule proposed by Friedlander and Iwaniec doesn't hold up for these specific, complex number towers when the parameters are tuned to a specific range.

The Story of the Stubborn Wave

Let's dive into the mechanics of this discovery. Imagine you are listening to a choir of singers, where each singer represents a number. In the Friedlander–Iwaniec world, the singers are arranged in a line, and they are supposed to sing a song that cancels itself out. The song has a specific rhythm, and the mathematicians believed that if you stopped the song at any point, the remaining sound would be barely a whisper.

Nguyen-Dang decided to test this by picking a very specific choir: the "divisor choir," where every singer is a number that can be built in many ways (like the number 12, which has many factor combinations). He set up a special experiment where he tuned the rhythm of the song to match the natural frequency of the singers.

Here is the trick: He chose a moment in time (a specific value for $x$ ) where the last few singers in the line were all hitting the exact same high note at the same time. This only works if the relationship between the total number of singers and the "zoom level" of the experiment follows a very specific rule (mathematically defined by a parameter $a$ ). Because the singers (the numbers) are all positive and the notes are all aligned, they didn't cancel out; they piled up! It's like a stadium wave where everyone stands up at the exact same second—the effect is huge, not tiny.

The paper shows that for these specific number towers (with $m \ge 4$ ), you can always find a moment where the last chunk of the song (the "endpoint") is perfectly synchronized, provided you choose the right parameters. When this happens, the sum of the numbers doesn't shrink to nothing. Instead, it grows to a size that is "power-sized." In math-speak, this means the value is proportional to a power of the number of singers, rather than being a tiny, negligible speck.

The author proves this with a clever mathematical construction. He shows that for any large number of singers $N$ , he can adjust the rhythm so that the last $H$ singers (where $H$ is a significant chunk of the total) are all singing in perfect harmony. Because the "volume" of each singer is at least 1, the total volume of this final block is massive. This massive block is the difference between two slightly different stopping points in the song. If the total song was supposed to be a whisper, how can a chunk of it be a roar? It can't. Therefore, the prediction that the whole song is a whisper must be wrong.

The Verdict

The paper delivers a clear, hard "no" to the specific prediction made by Friedlander and Iwaniec for these types of number towers when the tower has four or more layers ( $m \ge 4$ ) and the experiment is set up within a specific parameter range. It doesn't just suggest that the prediction might be slightly off; it constructs a mathematical proof that the prediction fails. The authors show that for the number towers built from $\zeta(s)^m$ (where $m$ is 4 or larger), there are always moments where the "dual sum" is large and powerful, defying the idea that it should be small. Specifically, while the counterexample for $m=4$ requires a careful setup, the paper proves that for $m=5$ and $m=6$ , the prediction fails even in the standard, printed range of numbers that mathematicians usually rely on.

This doesn't mean the whole field of number theory is broken. It just means that this one specific, elegant rule has a limit. It's like discovering that a bridge is perfectly safe for cars, but if you drive a heavy truck over it at a certain speed, it will shake violently. The bridge still works for cars, but the rule about "no shaking" isn't universal.

The paper is rigorous and definitive. It doesn't rely on computer simulations or guesses; it uses logical steps to prove that the "noise" in the sum is actually a "signal" that cannot be ignored. By finding these "counterexamples," the author forces mathematicians to rethink how they handle these complex sums. The dream of a single, simple rule that makes all the noise disappear for every type of number tower has been shattered, at least for this specific, important class of numbers. The journey to understand these numbers continues, but now we know that the path is more rugged and the waves are louder than we previously thought.

Technical Summary: Counterexamples of Friedlander–Iwaniec Dual Sums Conjecture

Problem Statement
The paper addresses a specific conjecture formulated by Friedlander and Iwaniec (2005) regarding the size of "sharply truncated nonlinear dual sums." In the context of automorphic $L$ -functions and Dirichlet series related by a functional equation, let $A(s)$ and $B(s)$ be dual series with analytic degree $m$ , conductor $D$ , and archimedean weight $k$ . Friedlander and Iwaniec defined the sum:
$B_{\ell,D}(x, N) := \sum_{n \le N} b(n) n^{-\beta_m} \cos\left( 2\pi m \left(\frac{nx}{D}\right)^{1/m} + \frac{\pi \ell}{4} \right)$
where $\beta_m = \frac{m+1}{2m}$ and $\ell = m - 3 - 2k$ .

Conjecture 1.1 predicts that for any $\varepsilon > 0$ , this sum satisfies the uniform bound:
$B_{\ell,D}(x, N) \ll_{\varepsilon, \kappa} (DNx)^{\varepsilon}$
uniformly in $x$ and $N$ (with fixed degree, conductor, and archimedean data). This conjecture suggests that the dual sum exhibits "subpower" cancellation, effectively replacing the trivial bound of roughly $N^{\theta_m}$ (where $\theta_m = \frac{m-1}{2m}$ ) with a bound that grows slower than any positive power of $N$ .

The paper investigates whether this conjecture holds for the specific case where $A(s) = B(s) = \zeta(s)^m$ , corresponding to the $m$ -fold divisor function $d_m(n)$ .

Methodology
The author employs a deterministic, local analysis of the phase of the oscillating sum, focusing on "endpoint resonance." The core mechanism relies on the observation that for a phase function $\Phi_x(n) = \kappa (nx)^{1/m}$ , the derivative $\Phi'_x(n)$ becomes small near a large endpoint $N$ when the parameter $x$ is scaled appropriately relative to $N$ .

Phase Alignment: The author constructs a sequence of parameters $x_N$ such that the phase of the final term ( $n=N$ ) aligns exactly with a maximum of the cosine function (or a point where the periodic profile $\Omega$ is non-zero).
Coherent Block Construction: By choosing $x_N$ such that the phase at $N$ is stationary (modulo $2\pi$ ), the author identifies a terminal interval of integers $[N-H_N, N]$ where the phase changes by less than a fixed constant. Within this interval, the cosine term maintains a uniformly positive real part.
Lower Bound Estimation: Since the coefficients $d_m(n)$ are strictly positive ( $d_m(n) \ge 1$ ), the sum over this coherent block does not cancel out. Instead, it accumulates to a magnitude proportional to the weighted mass of the block.
General Lemma: The paper first establishes a general "Sharp-Endpoint Sector Lemma" (Theorem 2.1) for sums of the form $\sum c_n n^{-\beta} \Omega(\kappa (nx)^\nu)$ , proving that if coefficients are bounded below, such sums must exhibit power-sized growth at specific endpoints.

Key Results
The paper provides explicit counterexamples to Conjecture 1.1 for $A(s) = \zeta(s)^m$ with $m \ge 4$ .

Theorem 1.2 (Sharp Endpoint Lower Bound): For any integer $m \ge 2$ and scaling parameter $0 < a < m-1$ , there exist sequences $x_N \asymp N^a$ and cutoffs $M_N \in \{N, N-H_N\}$ such that:
$|B_m(x_N, M_N)| \gg_{m,a,\lambda} N^{\Delta_m(a)}$
where the exponent is $\Delta_m(a) = \frac{m-3-2a}{2m}$ .
Corollary 1.3 (Failure of the Conjecture):
- Case (i): For $m \ge 4$ and $0 < a < \frac{m-3}{2}$ , the exponent $\Delta_m(a)$ is positive. Thus, the sum grows as a positive power of $N$ , directly contradicting the subpower bound $(DNx)^\varepsilon$ predicted by Conjecture 1.1.
- Case (ii): For $m \ge 6$ , the counterexample holds even within the range $1 \le N \le x$ (specifically with $a=1$ ), which is the range typically considered in applications.
- Case (iii): For $m \ge 5$ , counterexamples exist where $N \asymp x^{(m-1)/2}$ .

Significance and Claims
The paper claims to invalidate the specific route proposed by Friedlander and Iwaniec for bounding error terms in divisor problems via uniform dual sum estimates.

Refutation of Uniformity: The results demonstrate that the "subpower" bound in Conjecture 1.1 is false for the $m$ -fold divisor function when $m \ge 4$ . The conjecture fails because the functional equation is involutory; applying duality twice returns the original series, preventing the method from improving the trivial bound for the dual sum in general.
Impact on Conjecture 2: The paper notes that while these counterexamples invalidate the proposed "uniform sharp-dual-sum route" to Conjecture 1.6 (a bound on the error term $E_A(x)$ for the divisor problem), they do not disprove Conjecture 1.6 itself. The conjecture that $\Delta_m(x) \ll x^{(m-1)/(2m)+\varepsilon}$ (the Piltz divisor conjecture) remains open.
Mechanism Identification: The work identifies "endpoint resonance" as a fundamental obstruction to uniform subpower estimates in nonlinear dual sums. It clarifies that while smoothed sums or specific parameter ranges might allow for cancellation, the sharply truncated sums can exhibit large, coherent contributions at specific endpoints.

Conclusion
The paper concludes that Conjecture 1.1 of Friedlander and Iwaniec is false for the datum $\zeta(s)^m$ with $m \ge 4$ . The failure arises from the ability to align the oscillatory phase with the positive coefficients of the divisor function over a terminal block of integers, creating a "coherent block" with power-sized magnitude that cannot be suppressed by the functional equation alone.

The Story of the Stubborn Wave

The Verdict

Technical Summary: Counterexamples of Friedlander–Iwaniec Dual Sums Conjecture

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