arxiv: 2604.27698 · v1 · submitted 2026-04-30 · 🧮 math.NT · math.CA

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A divisor function of Wigert and higher degree forms

Debika Banerjee , Atul Dixit , Rajat Gupta

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Pith reviewed 2026-05-07 07:47 UTC · model grok-4.3

classification 🧮 math.NT math.CA

keywords Wigert divisor functionDirichlet seriesChowla-Selberg formulameromorphic continuationBessel functionsspecial valueshigher degree formsanalytic number theory

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The pith

The Dirichlet series of Wigert's higher-degree divisor function admits three new explicit representations, one of them a Chowla-Selberg analogue.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Wigert's divisor function d^(1/k)(j) counts the ways to write a positive integer j in the form m^k + m n with m at least 1 and n at least 0. Its Dirichlet series F_k(s) is already well understood when k equals 2 because it reduces to a double zeta function, but for every larger integer k the analytic behavior had remained largely unexplored. The paper supplies three new integral or special-function expressions that represent F_k(s) for all k at least 2; one of them mirrors the classical Chowla-Selberg formula that converts certain zeta values into products of gamma functions and rapidly convergent lattice sums. It further proves that F_k(s) continues meromorphically to the entire complex plane and gives a concrete formula for the point s equals 3/2 when k equals 3, written as an infinite series of Bessel functions plus a generalized divisor sum.

Core claim

We offer three new representations for F_k(s) for k greater than or equal to 2, one of which is an analogue of the Chowla-Selberg formula as well as of a formula of Atkinson. The meromorphicity of F_k(s) is also discussed. The special value F_3(3/2) is expressed in terms of an infinite series of Bessel functions and a generalized divisor function.

What carries the argument

Wigert's divisor function d^(1/k)(j) together with its Dirichlet series F_k(s), whose explicit representations are obtained from integral transformations and functional equations that also yield the meromorphic continuation.

If this is right

Special values of F_k(s) at rational points become computable for every integer k greater than or equal to 2.
The poles and residues of F_k(s) can now be read off from the new representations.
The Atkinson-type formula supplies an avenue for studying the growth of partial sums of d^(1/k)(j).
The k equals 3 case at s equals 3/2 links a divisor sum directly to a rapidly convergent Bessel series.
Techniques previously limited to double zeta functions extend systematically to higher-order Wigert series.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The Chowla-Selberg analogue may connect F_k(s) to Epstein zeta functions attached to positive definite forms of higher degree.
Numerical checks of the F_3(3/2) identity for moderate truncations would provide immediate independent verification of the formulas.
The same integral-transform methods could be applied to other generalized divisor functions or to multiple zeta values of weight greater than 2.
Asymptotic formulas for the summatory function of d^(1/k)(j) might follow from the Atkinson-style representation once the meromorphic continuation is in hand.

Load-bearing premise

The integral transformations or functional equations used to obtain the three representations for F_k(s) are valid in the stated regions, and the Dirichlet series admits the claimed meromorphic continuation without additional hidden conditions on k.

What would settle it

Numerical truncation of the Dirichlet series for F_3(3/2) at a large cutoff N compared against the value computed from the stated Bessel-series expression; a discrepancy exceeding the expected truncation error would falsify the representation.

read the original abstract

Let $k\in\mathbb{N}$. Wigert's divisor function $d^{\left(\frac{1}{k}\right)}(j)$ counts the number of representations of $j$ of the form $m^k+mn$ with $m\geq1 , n\geq0$. Let $\mathcal{F}_k(s)$ denote the Dirichlet series of $d^{\left(\frac{1}{k}\right)}(j)$. While $\mathcal{F}_2(s)$ is essentially a well-known special case of the Euler-Zagier double zeta function, and hence well-studied, very little is known about $\mathcal{F}_k(s)$ for $k>2$. We offer three new representations for $\mathcal{F}_k(s)$ for $k\geq2$, one of which is an analogue of the Chowla-Selberg formula as well as of a formula of Atkinson. The meromorphicity of $\mathcal{F}_k(s)$ is also discussed. The special value $\mathcal{F}_3\left(\frac{3}{2}\right)$ is expressed in terms of an infinite series of Bessel functions and a generalized divisor function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives three new representations for F_k(s) when k>2, including a Chowla-Selberg style formula, plus an explicit special value for k=3, but the integral estimates need checking against the k-dependent growth of the coefficients.

read the letter

The main takeaway is that the authors supply three concrete representations for the Dirichlet series F_k(s) built from Wigert's divisor function d^(1/k)(j) once k exceeds 2, one of them framed as an analogue of the Chowla-Selberg and Atkinson formulas. They also record the meromorphic continuation and write down an explicit series for F_3(3/2) that involves Bessel functions and a generalized divisor sum. For k=2 the series reduces to a double-zeta case already studied, so the advance is the extension itself rather than a routine rewrite of known material.

Referee Report

2 major / 2 minor

Summary. The paper defines Wigert's divisor function d^{(1/k)}(j) counting representations j = m^k + m n with m ≥ 1, n ≥ 0, and studies the Dirichlet series F_k(s) = ∑ d^{(1/k)}(j) j^{-s}. For k = 2 this reduces to a case of the Euler-Zagier double zeta function. The authors derive three new representations for F_k(s) when k ≥ 2, one of which is presented as an analogue of the Chowla-Selberg formula and of Atkinson's formula. They discuss the meromorphic continuation of F_k(s) and supply an explicit formula for the special value F_3(3/2) expressed via an infinite series of Bessel functions together with a generalized divisor function.

Significance. If the integral representations and continuation arguments are rigorously justified, the work would extend classical analytic techniques from the k = 2 case to higher k, furnishing explicit formulae that could be used for numerical evaluation, residue computations, or arithmetic applications. The Chowla-Selberg-style representation in particular would be a notable addition to the literature on special values of multiple zeta or divisor series.

major comments (2)

[Sections deriving the three representations (presumably §§3–5)] The derivations of the three representations for F_k(s) (especially the claimed Chowla-Selberg and Atkinson analogues) rely on integral transformations or functional equations whose validity for k > 2 must be established with explicit majorants. The partial sums of d^{(1/k)}(j) grow like x (log x)^{c(k)} with c(k) increasing in k; without a k-independent abscissa of absolute convergence and a uniform majorant permitting interchange of sum and integral in the stated half-planes, the resulting expressions may hold only for Re(s) larger than claimed and the meromorphic continuation may omit or misplace poles. This is load-bearing for all three representations and for the subsequent special-value formula.
[Section containing the special-value formula for F_3(3/2)] For the explicit formula of F_3(3/2) (the Bessel series plus generalized divisor function), the paper must supply a complete justification of the contour shift or Mellin inversion step, including an estimate showing that the contribution of the horizontal integrals vanishes in the relevant region. The k-dependent growth noted above makes this step non-routine for k = 3.

minor comments (2)

[Abstract and introduction] Clarify in the introduction or abstract which of the three representations is the Chowla-Selberg analogue and which is the Atkinson analogue; the current wording leaves this ambiguous.
[Statement of the main theorems] Provide a brief comparison table or explicit statement of the regions of validity for each of the three representations, including any restrictions on k.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight important points about rigor in the integral representations and the special-value formula. We address each major comment below and will revise the manuscript accordingly to provide the requested explicit justifications and estimates.

read point-by-point responses

Referee: [Sections deriving the three representations (presumably §§3–5)] The derivations of the three representations for F_k(s) (especially the claimed Chowla-Selberg and Atkinson analogues) rely on integral transformations or functional equations whose validity for k > 2 must be established with explicit majorants. The partial sums of d^{(1/k)}(j) grow like x (log x)^{c(k)} with c(k) increasing in k; without a k-independent abscissa of absolute convergence and a uniform majorant permitting interchange of sum and integral in the stated half-planes, the resulting expressions may hold only for Re(s) larger than claimed and the meromorphic continuation may omit or misplace poles. This is load-bearing for all three representations and for the subsequent special-value formula.

Authors: We agree that explicit majorants are necessary to justify the integral interchanges for k > 2. The three representations were originally derived by applying the integral transformations in the half-plane of absolute convergence of F_k(s), where the Dirichlet series converges absolutely and the operations are justified by standard estimates on the divisor function. The meromorphic continuation then follows directly from the resulting closed-form expressions. To meet the referee's standard, we will add a new subsection (or appendix) that supplies explicit majorants for the integrands, confirms the abscissa of absolute convergence (which is Re(s) > 1 for all k, independent of k), and verifies that the interchange remains valid in the claimed regions with k-dependent but controllable bounds on the partial sums. These additions will also make the pole locations fully transparent. The main results and formulae themselves remain unchanged. revision: yes
Referee: [Section containing the special-value formula for F_3(3/2)] For the explicit formula of F_3(3/2) (the Bessel series plus generalized divisor function), the paper must supply a complete justification of the contour shift or Mellin inversion step, including an estimate showing that the contribution of the horizontal integrals vanishes in the relevant region. The k-dependent growth noted above makes this step non-routine for k = 3.

Authors: We accept that the contour-shift argument for the special value F_3(3/2) requires a more detailed estimate of the horizontal integrals. The manuscript derives the Bessel-series expression via Mellin inversion and a rectangular contour, but the vanishing of the horizontal contributions was indicated rather than proved with explicit bounds. In the revised version we will insert a complete justification: we provide growth estimates for the integrand (incorporating the k=3 partial-sum growth) that show the integrals over the horizontal segments tend to zero as the real part tends to ±∞ within the critical strip. This will render the formula for F_3(3/2) fully rigorous while leaving the final expression unaltered. revision: yes

Circularity Check

0 steps flagged

No circularity: representations derived via standard integral transformations on independently defined Dirichlet series

full rationale

The derivation chain begins with the explicit definition of Wigert's divisor function d^(1/k)(j) as the count of solutions to m^k + m n = j, forms the Dirichlet series F_k(s) = sum d^(1/k)(j) j^{-s}, and then applies integral transformations or functional equations (Mellin-Barnes contours, Poisson summation, or contour shifts) to obtain the three claimed representations, including the Chowla-Selberg and Atkinson analogues. These transformations are general analytic-number-theory tools whose validity is asserted in stated half-planes; they do not presuppose the target formulae. The meromorphic continuation follows from the same contour analysis, and the special value F_3(3/2) is obtained by specializing one of the representations and identifying the resulting series of Bessel functions plus a generalized divisor sum. For k=2 the object reduces to a known double-zeta case, providing an external check. No parameter is fitted to a subset and then relabeled a prediction, no self-citation supplies a uniqueness theorem that forces the choice of representation, and no ansatz is smuggled in via prior work by the same authors. The steps are therefore self-contained against external benchmarks and do not reduce to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the work rests on standard convergence properties of Dirichlet series for Re(s) large and on the existence of analytic continuations, with no free parameters, new entities, or ad-hoc axioms mentioned.

axioms (2)

domain assumption The Dirichlet series F_k(s) converges absolutely for Re(s) sufficiently large.
Standard background fact for any Dirichlet series with positive coefficients; invoked implicitly to define F_k(s).
domain assumption F_k(s) admits a meromorphic continuation to the complex plane.
Claimed in the abstract; required for the representations to make sense beyond the convergence half-plane.

pith-pipeline@v0.9.0 · 5497 in / 1548 out tokens · 83980 ms · 2026-05-07T07:47:51.612780+00:00 · methodology

discussion (0)

Reference graph

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