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Almost primes in short intervals and averages of long Dirichlet polynomial approximations to $(\zeta'/\zeta)^{k}$

This paper establishes a connection between the mean square of long Dirichlet polynomial approximations to $(\zeta'/\zeta)^k$ and the short-interval variance of the convoluted von Mangoldt function $\Lambda^{*k}$ , proving under the Riemann Hypothesis and shifted moments conjectures that the leading coefficient of this variance matches a specific unitary trace-sum average derived from function field analogues.

Original authors: Siegfred Baluyot, John Brian Conrey, Ayesha Irfan, Nina C. Snaith

Published 2026-11-04

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CC BY 4.0

Original authors: Siegfred Baluyot, John Brian Conrey, Ayesha Irfan, Nina C. Snaith

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

In the vast landscape of mathematics, there is a region dedicated to understanding the hidden order within the sequence of prime numbers. These are the building blocks of all whole numbers, yet they appear to be scattered with a chaotic randomness that has puzzled thinkers for centuries. To make sense of this chaos, mathematicians often turn to a powerful tool called the Riemann zeta function. You can think of this function as a complex machine that takes a number as input and produces a new value, but its true power lies in its behavior along a specific, invisible line in the mathematical universe. The points where this machine outputs zero are like the fingerprints of the prime numbers; their arrangement holds the key to predicting how primes are distributed. For over a century, the most famous unsolved problem in mathematics, the Riemann Hypothesis, has claimed that all these fingerprints lie exactly on that central line. If this hypothesis is true, it would confirm a deep, underlying rhythm to the primes, though the rhythm is so subtle that it is easily drowned out by noise.

To study this rhythm, researchers often look at the "variance," which is a measure of how much the distribution of primes fluctuates in short stretches. Imagine counting how many primes appear in a small window of numbers, then moving that window forward and counting again. If the primes were perfectly evenly spaced, the count would never change. But they are not; the count jumps up and down. The size of these jumps, or the variance, tells mathematicians how wild the distribution is. A related concept involves "almost primes," which are numbers made by multiplying a specific number of prime factors together. By studying the fluctuations of these almost primes in short intervals, mathematicians can test theories about the deeper structure of the number system. This paper by S. Baluyot, J.B. Conrey, A. Irfan, and N.C. Snaith tackles the problem of calculating these fluctuations for complex, higher-order versions of almost primes, connecting them to the behavior of the Riemann zeta function's derivatives.

The authors set out to solve a specific puzzle: how does the average size of these fluctuations change as the length of the interval varies? They focused on a scenario where the interval is short compared to the total range of numbers being considered. To do this, they employed a technique that involves approximating the infinite series of numbers with a long, finite list, known as a Dirichlet polynomial. This allows them to work with manageable sums while still capturing the essential behavior of the infinite system. Their work relies on two major assumptions: the Riemann Hypothesis, which places the fingerprints of the primes on the central line, and a specific conjecture about the "moments" of the zeta function's logarithmic derivative. Moments, in this context, are a way of measuring the average power of the fluctuations, similar to how one might measure the average energy of a wave.

Under these assumptions, the researchers proved a striking relationship between two seemingly different quantities. On one side, they looked at the average size of the fluctuations of the almost primes in short intervals. On the other side, they examined the average size of the values of the zeta function's derivatives on the critical line. They found that the leading coefficient, which determines the scale of the fluctuations for the almost primes, is exactly the mathematical derivative of the leading coefficient for the zeta function's moments. In simpler terms, the rate at which the average energy of the zeta function changes as you adjust a parameter is precisely the same as the magnitude of the variance in the distribution of the almost primes. This connection was not just a guess; the authors derived it rigorously, showing that the complex formulas governing the zeta function directly dictate the statistical behavior of the primes in short intervals.

The paper also explores a parallel world of mathematics known as function fields, where numbers are replaced by polynomials over finite fields. In this setting, the researchers had previously studied similar variance problems using tools from random matrix theory, which treats the zeros of the zeta function as if they were the eigenvalues of a random unitary matrix. They found that in this polynomial world, the variance is governed by a specific coefficient derived from a matrix integral. The authors successfully bridged the gap between this polynomial world and the world of integers. They showed that the same coefficient that governs the variance in the polynomial world also appears as the leading term for the variance of the almost primes in the integers. This confirms that the statistical laws governing the distribution of primes in short intervals are consistent with the predictions made by random matrix theory, reinforcing the idea that the primes behave in a way that is statistically indistinguishable from a random process with specific constraints.

Furthermore, the team extended their findings to arithmetic progressions, which are sequences of numbers that leave the same remainder when divided by a fixed number. They investigated how the variance of the almost primes behaves when restricted to these specific sequences. Assuming the Generalized Riemann Hypothesis, which extends the original hypothesis to a broader class of functions, they demonstrated that the variance in these progressions also converges to the same leading coefficient found in the short-interval case. This result unifies the understanding of prime distribution across different contexts, showing that the underlying statistical structure remains robust whether one looks at a random short interval or a specific arithmetic sequence.

The authors' work is significant because it provides a precise formula for the leading term of these variances, a quantity that had been elusive for higher-order almost primes. While the results depend on the unproven Riemann Hypothesis and specific conjectures about moments, the internal logic of their derivation is sound and the connections they established are mathematically tight. They did not merely suggest a pattern; they proved that if the standard conjectures of the field are true, then the variance of the almost primes must follow this specific derivative relationship. This gives mathematicians a powerful new tool to predict the behavior of primes in short intervals and offers a deeper confirmation of the deep link between the analytic properties of the zeta function and the arithmetic properties of the integers. The paper stands as a testament to the power of connecting different areas of mathematics, using the language of random matrices and complex analysis to illuminate the ancient mystery of the prime numbers.

Technical Summary: Almost Primes in Short Intervals and Averages of Long Dirichlet Polynomial Approximations to $(\zeta'/\zeta)^k$

Problem Statement
This paper investigates the asymptotic behavior of the mean square of long truncations of the Dirichlet series for $(\zeta'/\zeta)^k$ on the critical line. The authors aim to establish a rigorous connection between this analytic problem and the short-interval variance of the convoluted von Mangoldt function $\Lambda^{*k}$ (where $\Lambda^{*k} = \Lambda * \cdots * \Lambda$ , $k$ times) over the integers. Specifically, the work seeks to determine the leading coefficients for:

The mean square of long Dirichlet polynomials approximating $(\zeta'/\zeta)^k$ .
The variance of sums of $\Lambda^{*k}$ in short intervals ( $H = X^{1-1/\alpha}$ ).
The variance of sums of $\Lambda^{*k}$ in arithmetic progressions.

These problems are motivated by function-field analogues analyzed using Random Matrix Theory (RMT), where similar variance problems correspond to unitary matrix integrals.

Methodology
The authors employ a combination of analytic number theory techniques and conjectures derived from RMT:

Conjectures on Shifted Moments: The analysis relies on Conjecture 1 (for the Riemann zeta function) and Conjecture 2 (for Dirichlet $L$ -functions). These conjectures provide precise asymptotic formulas for shifted moments of logarithmic derivatives, expressed as multiple contour integrals involving characteristic polynomials of unitary matrices.
Perron's Formula and Contour Shifting: The authors use truncated Perron formulas to relate sums of arithmetic functions to integrals of Dirichlet series. They carefully shift contours to isolate residue terms (main contributions) and bound error terms arising from zeros of $\zeta(s)$ and $L(s, \chi)$ , assuming the Riemann Hypothesis (RH) and Generalized Riemann Hypothesis (GRH).
Random Matrix Theory Analogues: The paper utilizes known results from RMT regarding averages of logarithmic derivatives of characteristic polynomials over the unitary group $U(N)$ . Specifically, it relates the variance of trace sums to unitary matrix integrals, denoted by $\delta_k(c)$ , where $c$ is the ratio of the total trace degree to the matrix size.
Contour Integral Decomposition: A significant portion of the technical work involves decomposing complex multiple contour integrals (arising from the moment conjectures) into sums over specific pole configurations. Lemmas 2.6, 2.7, and 2.8 are used to symmetrize, unsymmetrize, and bound these integrals, isolating the leading-order terms.

Key Contributions and Results

Derivative Relationship between Moments and Variances:
The central result is the proof that the leading coefficient of the short-interval variance of $\Lambda^{*k}$ is the derivative of the leading coefficient of the mean square of the corresponding long Dirichlet polynomial.
- Let $M_k(\alpha)$ be the leading coefficient for the mean square of the truncated series of $(\zeta'/\zeta)^k$ with cutoff $N = T^\alpha$ .
- Let $D_k(\alpha)$ be the leading coefficient for the short-interval variance of $\Lambda^{*k}$ with interval length $H = X^{1-1/\alpha}$ .
- Under Conjecture 1 and RH, the authors prove $D_k(\alpha) = M_k'(\alpha)$ . This mirrors a known result for divisor functions $d_k(n)$ established by Bettin and Conrey.
Explicit Formulas via Polynomials $P_{r,k}(\alpha)$ :
The authors derive an explicit piecewise polynomial form for $M_k(\alpha)$ (Theorem 2.2 and Theorem 2.5):
$M_k(\alpha) = \frac{1}{k!^2} \sum_{0 \le r < \alpha} \binom{k}{r}^2 P_{r,k}(\alpha)$
where $P_{r,k}(\alpha)$ is defined via a $2k$ -fold contour integral. Consequently, the variance coefficient is given by:
$D_k(\alpha) = \delta_k(\alpha) = \frac{1}{k!^2} \sum_{0 \le r < \alpha} \binom{k}{r}^2 P'_{r,k}(\alpha)$
The function $M_k(\alpha)$ is shown to be continuously differentiable $2r^2 - 1$ times at integer values $\alpha = r$ .
Arithmetic Progressions:
Under GRH and Conjecture 2, the paper establishes that the leading coefficient for the variance of $\Lambda^{*k}$ in arithmetic progressions, denoted $Q_k(\alpha)$ , is identical to the short-interval variance coefficient:
$Q_k(\alpha) = \delta_k(\alpha)$
This result holds for $k \ge 4$ under the stated conjectures, with the possibility of extension to $k=1,2,3$ if stronger error decay is assumed.
Function-Field Correspondence:
The paper recovers the predicted leading coefficients for integer statistics by showing they match the unitary trace-sum averages $\delta_k(c)$ derived from function-field models (specifically the work of Kuperberg and Lalín). This confirms that the "unitary trace-sum average" governs the variance of sums of $\Lambda^{*k}$ over monic polynomials in $\mathbb{F}_q[T]$ and, under the conjectures, over the integers.

Significance and Claims
The paper claims to provide a unified framework connecting the moments of logarithmic derivatives of $L$ -functions to the variance of almost-prime statistics in short intervals and arithmetic progressions.

It validates the heuristic that the variance of sums of coefficients of $(\zeta'/\zeta)^k$ is governed by the derivative of the mean-square moments, extending previous work on divisor sums.
It explicitly identifies the leading coefficients as piecewise polynomials derived from random matrix theory integrals, filling a gap in the literature where such explicit descriptions were unavailable for $k \ge 2$ .
The work serves as a bridge between the "swap" contributions in finite- $N$ moments (studied in forthcoming work [2]) and the asymptotic behavior of $L$ -function moments and arithmetic variances.

The authors maintain that their results are conditional on the Riemann Hypothesis, the Generalized Riemann Hypothesis, and specific shifted moments conjectures (Conjectures 1 and 2), which are motivated by but not yet proven by Random Matrix Theory. The paper does not claim to prove these conjectures but rather demonstrates the consequences of assuming them.

Technical Summary: Almost Primes in Short Intervals and Averages of Long Dirichlet Polynomial Approximations to (ζ′/ζ)k(\zeta'/\zeta)^k(ζ′/ζ)k

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Technical Summary: Almost Primes in Short Intervals and Averages of Long Dirichlet Polynomial Approximations to $(\zeta'/\zeta)^k$