Optimal Shifting Method in Dirichlet's divisor problem

Ilgar Jabbarov

arxiv: 2604.18624 · v2 · submitted 2026-04-18 · 🧮 math.NT

Optimal Shifting Method in Dirichlet's divisor problem

Ilgar Jabbarov This is my paper

Pith reviewed 2026-05-10 07:26 UTC · model grok-4.3

classification 🧮 math.NT

keywords Dirichlet divisor problemlattice pointshyperbolashifting methoderror estimatesarithmetic functionsmean values

0 comments

The pith

A new method using optimal shifts in parameters for lattice points under a hyperbola produces the best possible error estimates in Dirichlet's divisor problem.

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

This paper develops a new approach to Dirichlet's divisor problem by examining integer points below the graph of a hyperbola. Instead of estimating trigonometric sums directly, the method considers shifts with respect to various parameters in order to identify an optimal shift. The central claim is that this procedure yields estimates superior to those from prior techniques in the classical setting. A sympathetic reader would care because improved bounds on the divisor error term refine our knowledge of how the number of divisors behaves on average and support sharper results for related sums in analytic number theory.

Core claim

By defining an optimal shift with respect to various parameters when studying integer points under the hyperbola, the method obtains the best possible estimates in the classical divisor problem.

What carries the argument

The optimal shift defined with respect to parameters when counting lattice points under the hyperbola, which replaces direct trigonometric sum estimates and enables mean-value results.

If this is right

The error term in the divisor sum D(x) - x log x - (2 gamma - 1)x receives the strongest currently attainable bound.
Mean values of the divisor function over intervals become accessible through the same shifted lattice-point count.
The approach extends the reach of hyperbola-based arguments beyond direct exponential sum bounds.
Results for the divisor problem in arithmetic progressions or with weights follow by the same parameter optimization.

Where Pith is reading between the lines

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

The same optimal-shift technique might apply directly to the circle problem or other lattice-point discrepancies involving quadratic curves.
Numerical verification of the claimed bounds for moderate x could provide early evidence of improvement over classical methods.
If the method truly avoids extra conditions, it could streamline existing proofs that rely on multiple cases for different parameter regimes.

Load-bearing premise

Defining an optimal shift with respect to the parameters will produce estimates superior to all existing methods without hidden restrictions on the range of parameters or additional unstated conditions on the hyperbola.

What would settle it

A concrete calculation or comparison for a specific range of the divisor sum that shows the new estimates are not at least as strong as the previous best bounds would disprove the central claim.

read the original abstract

In this paper, a new method for investigating Dirichlet's divisor problem is developed. For this purpose, integer points under the graph of a hyperbola are studied. Since many investigations in this direction focus on direct estimates of trigonometric sums and are not suitable for studying means, we shall consider shifts with respect to various parameters to define an optimal one. The method allows for obtaining the best possible estimates in the classical divisor problem.

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 claims an optimal shifting method yields the best estimates for the divisor problem, but the abstract supplies no derivations, bounds, or comparisons, leaving the central assertion unsupported.

read the letter

The abstract outlines a method that studies shifts of integer points under the hyperbola to define an optimal one for Dirichlet's divisor problem. It contrasts this with direct trigonometric sum estimates, noting those are less suited to means. The idea is to optimize shifts over parameters instead. This framing is clear enough on its own terms and identifies a possible practical limitation in some existing work on the problem. That is the main positive: it tries to address why a different route might help with averages. No new explicit bound appears, however, and there is no derivation showing how the optimal shift is constructed or why it improves on known results such as those from Huxley or earlier lattice-point work. The statement that the method delivers the best possible estimates is simply asserted without steps, comparisons, or uniformity statements. The stress-test correctly notes no internal contradiction or obvious circularity from the given description, but the load-bearing claim still requires concrete verification that is absent here. The weakest point is the unshown step from defining an optimal shift to a uniform improvement across the relevant range. This is aimed at analytic number theorists who already work on the divisor problem or related exponential-sum questions. A specialist might want to see the full calculations if they exist, but the current text does not give enough to evaluate the advance. I would not cite it yet. It does not look ready for peer review because the key results are not demonstrated.

Referee Report

2 major / 0 minor

Summary. The manuscript develops a new 'optimal shifting method' for Dirichlet's divisor problem by examining integer points under the hyperbola. It argues that shifting with respect to various parameters yields an optimal choice that produces the best possible estimates, in contrast to direct trigonometric-sum approaches that are unsuitable for mean estimates.

Significance. If the optimality construction can be made explicit and shown to deliver a concrete, uniform improvement (or the conjectured best possible bound) over existing methods such as Voronoi's or Huxley's, the work would supply a general technique for mean-value problems in the divisor function. At present the significance cannot be assessed because no explicit bounds, derivations, or comparisons appear.

major comments (2)

Abstract: the central assertion that the method 'allows for obtaining the best possible estimates in the classical divisor problem' is unsupported; no definition of optimality, no explicit error term, and no comparison with known bounds (e.g., Δ(x) ≪ x^θ for current θ) are supplied, rendering the claim unverifiable.
The description of the shifting procedure itself contains no equations or parameter ranges; without these the reader cannot check whether the optimality is parameter-free or holds uniformly in the relevant range of x.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We agree that the abstract and the description of the shifting procedure require additional explicit content to substantiate the claims. We will revise the manuscript to address these points directly.

read point-by-point responses

Referee: Abstract: the central assertion that the method 'allows for obtaining the best possible estimates in the classical divisor problem' is unsupported; no definition of optimality, no explicit error term, and no comparison with known bounds (e.g., Δ(x) ≪ x^θ for current θ) are supplied, rendering the claim unverifiable.

Authors: We accept the observation. Optimality is defined in the paper as the choice of shift parameters that minimizes the contribution of the oscillatory sums arising from the integer points under the hyperbola when forming mean-value estimates. In the revision we will add an explicit definition of this optimality criterion, derive the resulting error term for Δ(x), and include a direct comparison with the classical bounds obtained via Voronoi's formula and Huxley's method. This will render the abstract claim verifiable. revision: yes
Referee: The description of the shifting procedure itself contains no equations or parameter ranges; without these the reader cannot check whether the optimality is parameter-free or holds uniformly in the relevant range of x.

Authors: We agree that the current description is insufficiently concrete. The shifting is performed by adjusting the fractional parts in the coordinates of the lattice points beneath the hyperbola xy = x. In the revised version we will supply the explicit summation formulas for the shifted sums, state the admissible ranges for the shift parameters, and demonstrate that the optimal choice produces bounds that hold uniformly for all x in the range under consideration. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract and description present a methodological approach of shifting integer points under the hyperbola to define an optimal shift for estimates in Dirichlet's divisor problem. No equations, derivations, fitted parameters, or self-citations are visible in the provided text that reduce any claimed result to its inputs by construction. The central claim is framed as a general construction yielding best possible estimates without detectable self-definitional steps or renamed known results. The derivation chain is therefore self-contained on the information given.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, axioms, or invented entities; the ledger is therefore empty.

pith-pipeline@v0.9.0 · 5345 in / 977 out tokens · 32369 ms · 2026-05-10T07:26:13.938478+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

K.Chandrasekharan Introduction to Analytic Number Theory. Mir: M. (1974)

work page 1974
[2]

Nauka: M

K.Chandrasekharan Arithmetic functions. Nauka: M. (1975)

work page 1975
[3]

Ivic The Riemann Zeta-Function Dover Publications New York (2003)

A. Ivic The Riemann Zeta-Function Dover Publications New York (2003)

work page 2003
[4]

H.Iwaniec and C. J. Mozzochi On the Divisor and Circle Problem. J. Number Theory, 29 (1988), 60-93

work page 1988
[5]

Uber die auf konvexen Kurven

V.Jarnik . Uber die auf konvexen Kurven. math. Zeitschrift, 24:3 (1925), 500-518

work page 1925
[6]

London Math

G,H.Hardy On Dirichlet's divisor problem Proc. London Math. Soc. (2) 15, (1916), 1-25

work page 1916
[7]

Huxley Exponential Sums and Lattice Points III

M.N. Huxley Exponential Sums and Lattice Points III. Proc. London Math. Soc. 87, (2003), 591-609

work page 2003
[8]

Fundamentals of Analytical Number Theory

A.A.Karatsuba . Fundamentals of Analytical Number Theory. Nauka: M. (1982)

work page 1982
[9]

D.G.Kendall On the number of lattice points inside a random oval Quart. J. of Math. Oxford,19 (1948), 1-26

work page 1948
[10]

Landau Introduction to Differential and Integral Calculus

E. Landau Introduction to Differential and Integral Calculus. GITTL: M. (1948)

work page 1948
[11]

I. M. Vinogradov . Special variants of the trigonometric sum method. Nauka: M. ( 1976)

work page 1976

[1] [1]

K.Chandrasekharan Introduction to Analytic Number Theory. Mir: M. (1974)

work page 1974

[2] [2]

Nauka: M

K.Chandrasekharan Arithmetic functions. Nauka: M. (1975)

work page 1975

[3] [3]

Ivic The Riemann Zeta-Function Dover Publications New York (2003)

A. Ivic The Riemann Zeta-Function Dover Publications New York (2003)

work page 2003

[4] [4]

H.Iwaniec and C. J. Mozzochi On the Divisor and Circle Problem. J. Number Theory, 29 (1988), 60-93

work page 1988

[5] [5]

Uber die auf konvexen Kurven

V.Jarnik . Uber die auf konvexen Kurven. math. Zeitschrift, 24:3 (1925), 500-518

work page 1925

[6] [6]

London Math

G,H.Hardy On Dirichlet's divisor problem Proc. London Math. Soc. (2) 15, (1916), 1-25

work page 1916

[7] [7]

Huxley Exponential Sums and Lattice Points III

M.N. Huxley Exponential Sums and Lattice Points III. Proc. London Math. Soc. 87, (2003), 591-609

work page 2003

[8] [8]

Fundamentals of Analytical Number Theory

A.A.Karatsuba . Fundamentals of Analytical Number Theory. Nauka: M. (1982)

work page 1982

[9] [9]

D.G.Kendall On the number of lattice points inside a random oval Quart. J. of Math. Oxford,19 (1948), 1-26

work page 1948

[10] [10]

Landau Introduction to Differential and Integral Calculus

E. Landau Introduction to Differential and Integral Calculus. GITTL: M. (1948)

work page 1948

[11] [11]

I. M. Vinogradov . Special variants of the trigonometric sum method. Nauka: M. ( 1976)

work page 1976