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arxiv: 2604.05559 · v1 · submitted 2026-04-07 · 🧮 math.CA

Some analytic properties of the partial theta function

Vladimir Petrov Kostov This is my paper

Pith reviewed 2026-05-10 19:21 UTC · model grok-4.3

classification 🧮 math.CA
keywords partial theta functionzero setanalytic continuationpower seriesRamanujancomplex zeros
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The pith

The partial theta function's real and complex zeros are separated by a vertical line in the complex plane for any q in (-1, 1) excluding zero.

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

The paper establishes separation properties for the zeros of the partial theta function defined as the infinite sum with quadratic exponents in q. For each q between 0 and 1, a vertical line at some Re x = -a with a at least 5 places all real zeros to the left and all complex zeros to the right. The same kind of separation is shown for q between -1 and 0, along with specific bounds preventing real zeros in certain intervals near zero. A reader would care because these properties describe where the function crosses zero without computing all terms, which could simplify analysis of related series in q-calculus or special functions.

Core claim

For each q in (0,1), there exists a line Re x = -a with a ≥ 5 such that all real zeros of θ(q, .) lie to its left and all complex zeros to its right. A similar property is proved for q in (-1,0). For q in (0,1), there are no real zeros ≥ -6. For q in (-1,0), there are no negative zeros ≥ -2.4 and no positive zeros ≤ 2.4, except the smallest one.

What carries the argument

Analytic continuation of the power series θ(q,x) combined with zero-counting arguments in the specified intervals of q.

If this is right

  • All real zeros of the partial theta function are bounded from above in their real parts for q > 0.
  • Complex zeros cannot have real parts smaller than the separating value -a.
  • For negative q, the real zeros avoid an interval around zero except possibly the smallest positive one.

Where Pith is reading between the lines

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

  • The separation could allow for better approximations of the function in different regions of the complex plane.
  • Numerical verification for various q values might reveal tighter bounds than the ones proved.
  • This property may extend to understanding the asymptotic location of zeros as q approaches the boundaries 0 or 1.

Load-bearing premise

The analytic continuation of the series allows reliable zero counting without additional corrections for q close to 0 or 1.

What would settle it

Discovering a q in (0,1) with a real zero having real part at least -5 or a complex zero with real part at most -5 would disprove the separation.

read the original abstract

We prove new properties of the zero set of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in (-1,0)\cup (0,1)$, $x\in \mathbb{R}$. We show that for each $q\in (0,1)$, there exists a line Re$x=-a$, $a\geq 5$, such that all real zeros of $\theta(q,.)$ lie to its left and all complex zeros to its right. A similar property is proved for $q\in (-1,0)$. For $q\in (0,1)$, there are no real zeros $\geq -6$. For $q\in (-1,0)$, there are no negative zeros $\geq -2.4$ and no positive zeros $\leq 2.4$, except the smallest one.

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 / 2 minor

Summary. The paper proves new zero-location properties for Ramanujan's partial theta function θ(q,x) := ∑_{j=0}^∞ q^{j(j+1)/2} x^j when q ∈ (-1,0) ∪ (0,1). For each fixed q ∈ (0,1) it establishes the existence of a vertical line Re x = -a with a ≥ 5 such that all real zeros lie to the left and all non-real zeros lie to the right; an analogous separation result is proved for q ∈ (-1,0). Explicit bounds are also given: no real zeros ≥ -6 when q > 0, and for q < 0 no negative zeros ≥ -2.4 together with no positive zeros ≤ 2.4 except the smallest one. The arguments rest on analytic continuation of the power series and zero-counting estimates in the complex plane.

Significance. If the derivations hold, the results supply concrete structural information on the zero set of the partial theta function, a classical q-series object with connections to partition theory and analytic number theory. The vertical-line separation between real and non-real zeros is a strong, falsifiable property that clarifies the geometry of the zero distribution and may facilitate further work on related q-functions or asymptotic questions. The explicit numerical bounds are immediately usable for numerical checks and theoretical extensions. The approach via analytic continuation and zero counting is standard in the field; the claims appear internally consistent with no circularity or free parameters.

minor comments (2)
  1. The abstract states the separation for q ∈ (-1,0) only as “a similar property”; spelling out the precise statement (including the value of a) would improve readability.
  2. A short paragraph recalling the known basic properties of θ(q,x) (e.g., its relation to the Jacobi theta function or prior zero-location results) would help place the new theorems in context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation of minor revision. The provided summary accurately reflects the main results of the paper concerning zero-location properties of the partial theta function.

Circularity Check

0 steps flagged

No circularity; proofs are direct from explicit series definition

full rationale

The paper defines θ(q,x) explicitly as the power series ∑ q^{j(j+1)/2} x^j and derives zero-location properties via analytic continuation and zero-counting in the complex plane. No fitted parameters, no predictions that reduce to inputs by construction, and no load-bearing self-citations or ansatzes are invoked for the central separation claims. The existence of the separating line Re x = -a (a ≥ 5) and the explicit bounds on real zeros follow from growth estimates and argument-principle applications that are independent of the target statements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard theorems from complex analysis applied to the power-series definition; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)
  • standard math Standard properties of power series convergence and analytic continuation in the complex plane
    Invoked to define θ(q,x) and discuss its zeros for q in the open intervals (-1,0) and (0,1).

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 1 internal anchor

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