Zsiflaw–Legeis Theorem

• The Zsiflaw–Legeis theorem is defined in two distinct contexts: one establishes asymptotic equality of positive and negative signum-areas of Hardy’s Z-function, while the other governs the equidistribution of reversed prime digits in arithmetic progressions.
• The signum-area formulation uses tailored disconnected sets and mean-value asymptotics to show that the integrated positive and negative excursions of Z(t) balance out as T tends to infinity.
• The reversed prime distribution version employs fixed-window analyses and additive combinatorial techniques to extend classical prime distribution results, with refined counting methods overcoming fixed-digit-length constraints.

The term Zsiflaw–Legeis Theorem is used in the supplied literature in two distinct ways. In an earlier analytic-number-theoretic usage, it denotes a law of asymptotic equality between positive and negative signum-areas of Hardy’s $Z$-function over specially constructed disconnected sets (Moser, 2013). In a later and now structurally different usage, it denotes a Siegel–Walfisz-type distribution theorem for digit reversals of primes in arithmetic progressions, first in fixed digit-length windows and then in a refined counting-by-size form (Bhowmik et al., 11 Jul 2025). The shared name is itself deliberate: in the prime-reversal literature, “Zsiflaw–Legeis” is “Siegel–Walfisz” reversed (Bhowmik et al., 11 Jul 2025).

1. Terminology and competing meanings

The nomenclature is context-dependent. The two principal meanings appearing in the record are summarized below.

Usage	Core object	Source
Signum-area law	Asymptotic equality of positive and negative areas under $Z(t)$ on $G_1(x)\cup G_2(x)$	(Moser, 2013)
Reversed-prime distribution law	Equidistribution of digit reversals of primes in arithmetic progressions	(Bhowmik et al., 11 Jul 2025)

In the Hardy $Z$-function setting, the name refers to the paper’s “law of asymptotic equality of signum-areas of $Z(t)$,” namely the asymptotic balance of areas above and below the $t$-axis over the disconnected sets $G_1(x)\cup G_2(x)$ (Moser, 2013). In the digital-reversal setting, the name is explicitly motivated by analogy with Siegel–Walfisz: the theorem gives a quantitative distribution law for reversed primes in residue classes, with fixed-window and absolute-counting variants (Bhowmik et al., 11 Jul 2025).

This suggests a genuine bifurcation of terminology rather than a single universally accepted theorem. In contemporary work on reversed primes, the phrase “Zsiflaw–Legeis” is used for arithmetic progression equidistribution of reversed primes, including a refined version adapted to additive problems (Harm et al., 21 May 2026).

2. The signum-area Zsiflaw–Legeis theorem for Hardy’s $Z$-function

In the 2013 usage, the ambient object is Hardy’s function

$$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$

where

$$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$

The paper works with Gram points $Z(t)$0 defined by $Z(t)$1, and shifted Gram points $Z(t)$2 defined by

$Z(t)$3

Using these, it introduces disconnected sets

$Z(t)$4

in the short-interval regime

$Z(t)$5

The associated positive and negative signum-areas over a measurable set $Z(t)$6 are

$Z(t)$7

The theorem itself states that, for $Z(t)$8, $Z(t)$9, and uniformly for $G_1(x)\cup G_2(x)$0,

$G_1(x)\cup G_2(x)$1

Equivalently,

$G_1(x)\cup G_2(x)$2

and in geometric notation

$G_1(x)\cup G_2(x)$3

The content is a balancing law: over the carefully tailored disconnected set $G_1(x)\cup G_2(x)$4, the positive and negative excursions of $G_1(x)\cup G_2(x)$5 contribute asymptotically equal area (Moser, 2013).

The theorem is built on mean-value asymptotics over the disconnected sets: $G_1(x)\cup G_2(x)$6 uniformly for $G_1(x)\cup G_2(x)$7. These formulas decompose a short Hardy–Littlewood integral into contributions over $G_1(x)\cup G_2(x)$8 and $G_1(x)\cup G_2(x)$9, and for $Z$0 recover

$Z$1

The signum-area theorem is therefore not an isolated geometric statement; it is the geometric reformulation of a fine mean-value theory for $Z$2 on disconnected sets (Moser, 2013).

3. The fixed-window Zsiflaw–Legeis theorem for reversals of primes

In the later literature, the theorem concerns the digital reverse of primes. For a fixed base $Z$3, every $Z$4 has a base-$Z$5 expansion

$Z$6

with digit length

$Z$7

The absolute digital reverse is

$Z$8

and the relative reverse in a fixed digit window of length $Z$9 is

$Z(t)$0

On the interval $Z(t)$1, one has $Z(t)$2.

The fixed-window counting function is

$Z(t)$3

The necessary congruence conditions are

$Z(t)$4

The density factor is

$Z(t)$5

if $Z(t)$6 and $Z(t)$7, and $Z(t)$8 otherwise.

The quantitative theorem states that for fixed $Z(t)$9 and $t$0,

$t$1

uniformly for

$t$2

where $t$3 and the implied constants depend only on $t$4 and are effectively computable (Bhowmik et al., 11 Jul 2025).

This is the reversed-digit analogue of a Siegel–Walfisz theorem. The same paper also proves a reversed-digit analogue of Dirichlet’s theorem, called “Telhcirid’s theorem on arithmetic progressions”: for $t$5, $t$6, $t$7, and $t$8, there are infinitely many primes $t$9 such that $G_1(x)\cup G_2(x)$0 (Bhowmik et al., 11 Jul 2025).

A significant aspect of the 2025 result is the removal of earlier base-size restrictions. Previous work had required $G_1(x)\cup G_2(x)$1, later improved to $G_1(x)\cup G_2(x)$2, whereas the arbitrary-base theorem holds for every integer base $G_1(x)\cup G_2(x)$3 (Bhowmik et al., 11 Jul 2025).

4. Weakly digital functions, proof architecture, and absolute-counting forms

The arbitrary-base theorem is proved by extending the Martin–Mauduit–Rivat framework from digital functions to weakly digital functions, which may depend on both digit and position. For a seed $G_1(x)\cup G_2(x)$4, the basic phase is

$G_1(x)\cup G_2(x)$5

Reversal is modeled by taking, for fixed $G_1(x)\cup G_2(x)$6 and $G_1(x)\cup G_2(x)$7,

$G_1(x)\cup G_2(x)$8

so that

$G_1(x)\cup G_2(x)$9

The key prime-exponential-sum theorem has the shape

$Z$0

where

$Z$1

Its proof combines product formulas for normalized sums $Z$2, pointwise $Z$3 bounds, discrete $Z$4 bounds, a hybrid large-sieve type estimate, and Type I and Type II bilinear bounds obtained through Vaughan’s identity (Bhowmik et al., 11 Jul 2025).

The distribution theorem for $Z$5 is then derived by additive characters. Major arcs correspond to the condition that the congruence collapses to an essential modulus $Z$6, while minor arcs are controlled through the weakly digital prime-exponential-sum estimate. This yields Siegel–Walfisz-type formulas for

$Z$7

uniformly for

$Z$8

with error

$Z$9

The same paper also gives a “pure” absolute version, counting primes up to $$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$0 by their actual reversal rather than inside a fixed window. If

$$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$1

then

$$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$2

uniformly for

$$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$3

Thus the fixed-window theorem and the absolute-counting theorem are two layers of the same analytic structure: the first isolates stable digit-length blocks, while the second sums those blocks into a global counting statement (Bhowmik et al., 11 Jul 2025).

5. The refined Zsiflaw–Legeis theorem without fixed digit length

The 2026 refinement reformulates the reversed-prime theorem in a form directly suited to additive problems. For a fixed base $$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$4, write

$$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$5

when

$$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$6

If $$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$7 is prime, its reversal is

$$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$8

The weighted progression count is

$$Z(t)=e^{i\theta(t)}\zeta\!\left(\tfrac12+it\right),$$9

The earlier fixed-$$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$0 theorem in this literature has the asymptotic shape

$$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$1

uniformly for

$$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$2

where

$$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$3

A modified effective variant restricts to reversals coprime to $$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$4 and yields the same main term and error, again for $$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$5 (Harm et al., 21 May 2026).

The paper’s main contribution is the refined Zsiflaw–Legeis theorem without fixing $$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$6. Let

$$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$7

Then, for any $$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$8, $$\theta(t)=\arg\Gamma\!\left(\tfrac14+\tfrac{it}{2}\right)-\tfrac{t}{2}\log\pi.$$9, $Z(t)$00, and $Z(t)$01,

$Z(t)$02

uniformly for

$Z(t)$03

Here $Z(t)$04 is ineffective because the argument uses Siegel–Walfisz for primes (Harm et al., 21 May 2026).

Three structural features are encoded directly in the formula. First, the leading-digit constraint is absorbed by $Z(t)$05. Second, the local obstruction modulo $Z(t)$06 appears through

$Z(t)$07

Third, the theorem is specifically designed to remove the fixed-digit-length restriction that obstructs circle-method applications (Harm et al., 21 May 2026).

6. Additive consequences, effectivity, and common misconceptions

The refined theorem feeds directly into a Hardy–Littlewood circle-method analysis of mixed prime/reversed-prime representation problems. The paper proves four headline consequences: every large odd integer is the sum of a prime and two reversed primes; every large odd integer is the sum of two primes and a reversed prime; almost all even integers are the sum of a prime and a reversed prime; and all large integers are the sum of a reversed prime and a square-free number (Harm et al., 21 May 2026).

More precisely, for the ternary problems one obtains asymptotics of the form

$Z(t)$08

and

$Z(t)$09

For the binary problem, all but

$Z(t)$10

even integers $Z(t)$11 are representable as

$Z(t)$12

with $Z(t)$13. For the square-free complement problem,

$Z(t)$14

which implies representations

$Z(t)$15

for all sufficiently large $Z(t)$16 (Harm et al., 21 May 2026).

Methodologically, the major arcs use the classical prime sum

$Z(t)$17

together with the reversed-prime sum

$Z(t)$18

and the refined Zsiflaw–Legeis theorem supplies the major-arc asymptotics for $Z(t)$19. Minor arcs use Vinogradov-type bounds for $Z(t)$20 and Parseval-based $Z(t)$21 control for $Z(t)$22 (Harm et al., 21 May 2026).

Two misconceptions are especially common. The first is that “Zsiflaw–Legeis theorem” names a single universally fixed result. The literature supplied here does not support that reading: one usage concerns signum-areas of Hardy’s $Z(t)$23-function, while another concerns arithmetic progression equidistribution of reversed primes (Moser, 2013). The second is that all versions have the same analytic status. The arbitrary-base fixed-window theorem is stated with effective constants depending only on $Z(t)$24 (Bhowmik et al., 11 Jul 2025), whereas the refined counting-by-$Z(t)$25 theorem is ineffective because it relies on Siegel–Walfisz for primes (Harm et al., 21 May 2026).

Taken together, these results show that the name now covers two unrelated but structurally parallel themes: a geometric balancing law for oscillatory values of $Z(t)$26, and a reversed-digit analogue of classical prime distribution theorems. The latter has become the central meaning in recent additive work on reversed primes (Harm et al., 21 May 2026).