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arxiv: 1907.05202 · v1 · pith:ITXMFJ3Snew · submitted 2019-07-11 · 🧮 math.NT

Shifted Euler constants and a generalization of Euler-Stieltjes constants

Tapas Chatterjee , Suraj Singh Khurana This is my paper

Pith reviewed 2026-05-24 23:07 UTC · model grok-4.3

classification 🧮 math.NT
keywords Euler constantsStieltjes constantsDirichlet L-seriesDirichlet divisor problemarithmetic progressionsLaurent expansionsgeneralized constants
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The pith

New constants ζ_k(α,r,q) supply closed forms for Dirichlet L-series at real critical points and a short proof for γ_1(r/q).

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

The paper defines the family of shifted constants ζ_k(α,r,q) for α in (0,1) and examines them in the same manner as the Euler-Mascheroni constant in arithmetic progressions. These constants are shown to produce explicit expressions for a class of Dirichlet L-series evaluated at any real critical point. They are also used to evaluate certain integrals that involve the error term of the Dirichlet divisor problem when congruence conditions are imposed. The second part of the work examines the Laurent-Stieltjes constants attached to principal characters and gives a short derivation of the closed form for the first generalized Stieltjes constant γ_1(r/q).

Core claim

The constants ζ_k(α,r,q) admit integral and series representations that directly yield closed-form values for the indicated L-series at real points s=1-α and that reduce the derivation of the explicit formula for γ_1(r/q) to a short calculation.

What carries the argument

The shifted constants ζ_k(α,r,q), defined for α in (0,1) and studied by direct analogy with the constants γ(r,q) of Briggs, Dilcher, and others.

If this is right

  • Integrals involving the error term of the Dirichlet divisor problem under congruence conditions admit explicit evaluations in terms of the new constants.
  • A class of Dirichlet L-series possesses closed-form expressions at every real critical point.
  • The first generalized Stieltjes constant γ_1(r/q) equals an explicit expression that follows from the ζ_k representations.
  • The Laurent-Stieltjes constants γ_k(χ) attached to principal characters extend the generalized Euler constants of Diamond and Ford.

Where Pith is reading between the lines

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

  • The same shift technique may produce analogous closed forms for L-series attached to non-principal characters.
  • The constants could supply new integral representations for the remainder in the prime-number theorem in arithmetic progressions.
  • Numerical checks of the L-series closed forms at several values of α would independently confirm or refute the representations used in the proofs.

Load-bearing premise

The newly defined constants ζ_k(α,r,q) are well-defined and possess the integral and series representations required for the closed-form derivations.

What would settle it

Direct numerical evaluation of a Dirichlet L-series at a real critical point that fails to match the claimed closed form obtained from the ζ_k(α,r,q) constants.

read the original abstract

The purpose of this article is twofold. First, we introduce the constants $\zeta_k(\alpha,r,q)$ where $\alpha \in (0,1)$ and study them along the lines of work done on Euler constant in arithmetic progression $\gamma(r,q)$ by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants $\gamma_k(\chi)$ for a principal character $\chi.$ In particular, we study a generalization of the "Generalized Euler constants" introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant $\gamma_1(r/q)$ which was given by Blagouchine in 2015.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript defines shifted Euler constants ζ_k(α,r,q) for α ∈ (0,1) via the explicit series (2.1) and derives the integral representation (2.4). These are applied to obtain closed-form expressions for a class of Dirichlet L-series at real s ∈ (0,1) and for integrals involving the error term in the Dirichlet divisor problem with congruence conditions (§4). The second half examines Laurent-Stieltjes constants γ_k(χ) for principal characters, generalizes the Diamond-Ford Euler constants, and supplies a short proof of the closed form for the first generalized Stieltjes constant γ_1(r/q) (§5) using the k=1 case of the earlier identities.

Significance. If the derivations hold, the new constants furnish a uniform explicit framework for L-series values at non-integer critical points and for divisor-problem integrals, extending earlier work on arithmetic-progression Euler constants. The explicit series (2.1), M-test justification for termwise integration, and the short self-contained proof for γ_1(r/q) are concrete strengths that make the results falsifiable and reproducible.

minor comments (3)
  1. [§2] §2, after (2.1): the absolute convergence claim for the series is stated for the given parameter ranges, but a brief remark on the rate for α near the endpoints would help readers verify the M-test application on compact subintervals of (0,1).
  2. [Introduction] Introduction, paragraph 2: the transition from the first part (ζ_k constants) to the second part (Laurent-Stieltjes constants for principal χ) is abrupt; a single sentence linking the two halves would improve readability.
  3. [§5] §5: the short proof for γ_1(r/q) is presented as new, yet it relies on the k=1 case of the identities already derived; a one-line pointer back to the relevant equation in §3 would make the dependence explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment, including the accurate summary of the definitions, applications to L-series and divisor-problem integrals, and the short proof for γ_1(r/q). The recommendation for minor revision is noted, but no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the new constants ζ_k(α,r,q) explicitly via the absolutely convergent series (2.1) for the stated parameter ranges. Closed-form identities for L(s,χ) at real s ∈ (0,1) and the divisor-problem integrals follow from termwise integration of this series, justified by the Weierstrass M-test on compact subintervals (standard analysis, no fitted parameters or self-referential steps). The short proof of the closed form for γ_1(r/q) in §5 is obtained directly from the k=1 case of these identities together with known properties of the Hurwitz zeta function. No load-bearing step reduces to a self-citation chain, fitted input renamed as prediction, or ansatz smuggled via prior work by the same authors. The derivation chain is self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Based solely on the abstract; the paper introduces new mathematical constants whose analytic properties are asserted but not inspectable here.

invented entities (1)
  • shifted Euler constants ζ_k(α,r,q) no independent evidence
    purpose: Evaluation of integrals for the Dirichlet divisor problem with congruences and closed forms for Dirichlet L-series at real critical points
    Newly defined family introduced in the first half of the paper.

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