Mathematical Pipeline Unlocks Hidden Patterns in Prime Number Frequencies

Researchers investigate the statistical properties of integer partitions, revealing deep connections between number theory and modular forms. Hartosh Singh Bal, alongside colleagues, demonstrate a novel method for analysing frequency moments of these partitions, utilising a transform that links them to explicit divisor sums. This approach allows the team to effectively detect and prove Ramanujan-type congruences, patterns within the coefficients of modular forms, offering a powerful tool for arithmetic investigation. The study not only recovers and certifies known congruences for ordinary partition moments, such as those for odd moments in nonzero residue classes, but also extends the analysis to overpartitions, uncovering contrasting behaviours and establishing new progressions, including a quadratic twist yielding a certified congruence.

Certifying prime modularity of partition frequency moments via divisor sums

Researchers have developed a new technique for detecting and proving Ramanujan-type congruences for frequency moments of partition statistics. This work centres on a transform that expresses moment generating functions as explicit divisor-sum series, revealing modular forms whose coefficients can be rigorously certified modulo primes using a Sturm bound.

The study builds upon the Partition, Frequency Enumeration framework, extending its capabilities to identify previously unknown congruences and structural properties within partition theory. This innovative pipeline offers a systematic approach to uncovering hidden arithmetic patterns in combinatorial generating functions.

The research successfully recovers and certifies several congruences for odd frequency moments in nonzero residue classes, such as M3(7n + 5) ≡ 0 (mod 7) and M3(11n + 6) ≡ 0 (mod 11). Applying the same methodology to overpartitions, the team certified a family of zero-class congruences, including those for m = 5, 7, 11, and 13.

Notably, a striking contrast emerged: no nonzero residue-class congruences were observed for overpartition moments within the scanned range. This suggests a fundamental difference in the arithmetic properties of ordinary partitions and overpartitions. Furthermore, the researchers demonstrated that filtering partition statistics using the Glaisher, character dictionary generates new Ramanujan-type progressions.

Specifically, a quadratic twist yields the certified congruence c Mχ5 3 (5n + 4) ≡ 0 (mod 5). This highlights the power of combining combinatorial filtering with modular form techniques to uncover deeper arithmetic structures. The methodology is not limited to ordinary partitions; it extends to any Euler-type product derived from η-products with known modular data, establishing a versatile “congruence detection machine”.

This work advances the partition-frequency program by unifying classical recursions, Ramanujan-type congruences, and quasimodularity under a single framework. The study’s findings have implications for understanding the arithmetic of partition functions and offer a powerful tool for exploring the interplay between number theory and combinatorics. The developed pipeline provides an effective means of both detecting and rigorously proving Ramanujan-type congruences, opening new avenues for research in this area.

Divisor convolution and the Partition Frequency Enumeration matrix for congruence certification

A transform expressing moment generating functions as times explicit divisor–sum series forms the basis of this work. Specifically, the research investigates frequency moments of partition statistics arising from Euler products, utilising a methodology centred around the Partition, Frequency Enumeration (PFE) matrix.

This matrix establishes a direct link between weighted frequency sums and divisor convolutions σm(n; χ), subsequently connecting these to Eisenstein series Em+1(τ, χ). The PFE matrix, an infinite upper-triangular matrix, is constructed with entries defined by aij = 1 if i divides j, and 0 otherwise, enabling the unrolling of frequency recursions.

Researchers then applied this pipeline to both ordinary partitions and overpartitions to certify congruences. For ordinary partitions, the study recovers and certifies congruences for odd moments in nonzero residue classes, demonstrating congruences for instances such as M2(5n+1) and M2(7n+3). Applying the same methodology to overpartitions revealed a contrasting landscape, with no nonzero residue–class congruences observed within the scanned range, and certification of zero–class congruences like Mm(ln) ≡0 (mod l).

Furthermore, the study demonstrates that filtering the statistic via the Glaisher–character dictionary creates new Ramanujan–type progressions. This involves establishing a complete dictionary between Glaisher’s divisor selections and Dirichlet twists σm(·; χ), allowing every filtered frequency moment to correspond to a modular form with computable level and Nebentypus, culminating in the certified congruence c Mχ5 3 (5n + 4) ≡0 (mod 5). The computational range extended to primes 5 ≤l≤97 and moments m ≤99, providing a robust framework for congruence detection.

Certified congruences for partition and overpartition moments via modular forms

Frequency moments of partition statistics were studied via a transform expressing moment generating functions as times explicit divisor–sum series. When the input is modular, specifically a –quotient, the research yields (quasi)modular forms whose coefficients can be projected to arithmetic progressions and certified modulo primes using a Sturm bound.

This provides an effective pipeline for detecting and proving Ramanujan–type congruences for frequency moments. For ordinary partitions, several congruences for odd moments in nonzero residue classes, such as and, were recovered and certified. Applying the same pipeline to overpartitions, a family of zero–class congruences, including, was certified.

This contrasts with the ordinary partition case, as no nonzero residue–class congruences were observed for overpartition moments within the scan range. Filtering the statistic via the Glaisher–character dictionary created new Ramanujan–type progressions, exemplified by a quadratic twist yielding the certified congruence .

The work demonstrates that congruences, twists, and irregularities acquire explicit combinatorial avatars in terms of parts of partitions. The frequency recursion, Fk(n) = p(n −k) + Fk(n −k), unrolls to Fk(n) = Σ j≥1 p(n −jk), with all negative arguments vanishing. The explicit rows for fixed n, denoted Pn:= p(n −1), p(n −2), p(n −3), …, show that each row samples Pn along the multiples of k.

These recursions are packaged into the infinite upper, triangular matrix A = (aij)i,j≥1, defined by aij = (1, i | j, 0, otherwise). The master identity establishes that weighted frequency sums correspond to divisor sums, specifically X i≥1 f(i) Fi(n) = n X d=1 σ(f)(d) p(n −d). For arithmetic weight f(i) = i, this yields the Euler, Ramanujan, Ford recursion.

For f(i) = μ(i), the Möbius function, the sum equates to p(n −1). Furthermore, for f(i) = im, the frequency moments Mm(n) := X k≥1 kmFk(n) = n X d=1 σm(d) p(n −d) are obtained. The canonical moments, σ(m) A (n) := X r|n c(r) rm, and MA m(n) := n X d=1 σ(m) A (d) b(n −d), are defined for an Euler-type product A(q) = qα Y r≥1 (1 −qr)c(r).

The master transform, X n≥0 MA f (n)qn = B(q) · X r≥1 f(r)c(r) qr 1 −qr, connects weighted frequency sums to divisor sums. Assuming an inverse-companion case B(q) = A(q)−1, the logarithmic derivative yields X n≥0 MA 1 (n) qn = q d dqB(q), implying MA 1 (n) = n b(n). Consequently, for every prime lone, MA 1 (ln) ≡0 (mod l) for n ≥0, and this congruence extends to all odd m ≡1 (mod l−1).

Partition congruences via divisor sums and modular forms

Researchers have established a systematic method for investigating frequency moments of partition statistics derived from Euler products, leveraging a transform that links moment generating functions to divisor–sum series. This approach, when applied to modular forms, yields (quasi)modular forms with coefficients amenable to analysis and certification, enabling the detection and verification of Ramanujan–type congruences for frequency moments.

Specifically, several congruences for odd moments of ordinary partitions in nonzero residue classes have been recovered and confirmed, such as those relating to powers of 691. Furthermore, the same pipeline was applied to overpartitions, revealing a distinct pattern; while zero–class congruences were identified, no nonzero residue–class congruences were observed within the examined range.

The application of the Glaisher–character dictionary as a filtering mechanism was also demonstrated to generate new Ramanujan–type progressions, including a certified quadratic twist congruence. These findings highlight the interplay between partition statistics, modular forms, and arithmetic properties of numbers, offering a refined understanding of Ramanujan-type congruences.

A limitation acknowledged is the computational scope of the search for nonzero residue–class congruences for overpartitions, which was limited to a specific range. Future research should focus on extending the analysis of moment generating functions by examining their decomposition into Hecke eigenforms and Shimura lifts, alongside investigating the associated L, values.

This detailed analysis promises to clarify the spectral manifestation of irregular primes and potentially broaden the application of partition statistics to explore the arithmetic of modular forms beyond the initial examples presented. The authors also provide a comprehensive Glaisher dictionary and detailed computational methods for verifying the congruences, facilitating further investigation in this area.