Turyn Polynomials and the Quarter-Shift

• Turyn polynomials are shifted quadratic-character polynomials derived from the Legendre symbol and form a special case of mixed character-sum families.
• Recent results prove that their normalized even L2k-norm profile is minimized at the quarter-shift (α = 1/4), confirming longtime conjectures in analytic number theory.
• These polynomials connect to Littlewood constructions and Mahler’s problem, offering practical insights into flatness phenomena and autocorrelation sequence applications.

Searching arXiv for the cited Turyn-polynomial papers and closely related terminology. Turyn polynomials are most explicitly realized as shifted quadratic-character polynomials obtained from the Legendre symbol, and in contemporary analytic number theory they are treated both as special cases of incomplete mixed character sums and as near-Littlewood polynomials whose extremal behavior is governed by the choice of shift. For an odd prime $p$, a standard form is

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$

where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol and $a/p\to\alpha$ is the normalized shift. In the broader framework of shifted character-sum polynomials, Turyn polynomials correspond to the case $\beta=1$ in

$$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$

so they sit naturally beside Fekete polynomials and other quadratic-character Littlewood-type constructions. Recent work establishes that, for every integer $k\ge 2$, the asymptotic normalized $L_{2k}$-norm profile of Turyn polynomials is minimized at the quarter-shift $\alpha=\tfrac14$, proving a conjecture of G\"unther and Schmidt (Bober et al., 7 Oct 2025). Closely related work on Mahler’s problem shows that the same quarter-shift regime yields normalized Mahler measure exceeding $0.95$ after the standard one-coefficient correction that produces genuine $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$0-coefficient Littlewood polynomials (Mossinghoff, 2024).

1. Definition and placement within character-sum polynomial families

In the notation used for incomplete mixed character sums, the central object is

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$1

for a primitive Dirichlet character $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$2, where $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$3. This can be viewed as the polynomial

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$4

evaluated on the unit circle, and it “naturally generaliz[es] the well-known variants of Fekete ($$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$5 and $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$6) and Turyn ($$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$7) polynomials.” In this sense, Turyn polynomials are the $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$8 subfamily of shifted mixed character-sum polynomials (Bober et al., 7 Oct 2025).

For quadratic characters, the relevant specialization is

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$9

where $\left(\frac{\cdot}{p}\right)$0 is the quadratic character modulo the prime $\left(\frac{\cdot}{p}\right)$1. The paper devoted to Mahler’s problem uses an equivalent cyclic-shift formulation based on the Fekete polynomial

$\left(\frac{\cdot}{p}\right)$2

and defines the Turyn polynomial by

$\left(\frac{\cdot}{p}\right)$3

with asymptotic shift parameter $\left(\frac{\cdot}{p}\right)$4 (Mossinghoff, 2024).

This family is therefore best understood as a shifted Legendre-symbol family. In the formulation

$\left(\frac{\cdot}{p}\right)$5

the parameter $\left(\frac{\cdot}{p}\right)$6 shifts the quadratic-character coefficients, and the normalized asymptotic location is $\left(\frac{\cdot}{p}\right)$7. The coefficients are essentially $\left(\frac{\cdot}{p}\right)$8, with the usual single zero arising from the Legendre symbol at the residue class $\left(\frac{\cdot}{p}\right)$9; after excluding or correcting that coefficient in the standard way, these become Littlewood polynomials in the usual sense (Bober et al., 7 Oct 2025).

A particularly important special case is the quarter-shift $a/p\to\alpha$0. The modern literature repeatedly singles this out, both in norm-minimization results and in Mahler-measure computations, as the distinguished Turyn regime (Bober et al., 7 Oct 2025).

2. Classical form, coefficient structure, and Littlewood-polynomial status

The explicit classical shifted Turyn polynomial appearing in recent analytic work is

$a/p\to\alpha$1

Here the Legendre symbol furnishes a quadratic-character coefficient sequence. Since $a/p\to\alpha$2, the unshifted Fekete polynomial has one zero coefficient and all other coefficients are $a/p\to\alpha$3; the same phenomenon persists under cyclic shift for $a/p\to\alpha$4, where exactly one coefficient is $a/p\to\alpha$5, namely at the index where $a/p\to\alpha$6 (Mossinghoff, 2024).

To obtain genuine Littlewood polynomials, one introduces the companion polynomials

$a/p\to\alpha$7

These have all coefficients equal to $a/p\to\alpha$8, and the asymptotic normalized $a/p\to\alpha$9-norms and Mahler measure are unchanged by this one-term modification in the sense made precise by the asymptotic results (Mossinghoff, 2024).

The terminology “Turyn polynomial” is therefore exact in one well-defined sense: it denotes a cyclically shifted Fekete polynomial, equivalently a shifted quadratic-character polynomial with $\beta=1$0. At the same time, the broader Turyn literature contains adjacent but distinct objects. Turyn-type sequences are quadruples $\beta=1$1 whose associated sequence polynomials satisfy

$\beta=1$2

which is a different construction, although closely related in autocorrelation theory and Hadamard-matrix applications (Best et al., 2012).

A separate source of terminological ambiguity arises from Turán expressions, defined for a polynomial sequence $\beta=1$3 by

$\beta=1$4

These belong to the theory of Turán inequalities and weak Hurwitz stability and are unrelated to Turyn polynomials except for the similarity of names (Chasse et al., 2014).

3. Asymptotic $\beta=1$5-norm theory and the quarter-shift theorem

The central asymptotic extremal theorem for Turyn polynomials concerns their normalized even norms. The $\beta=1$6-norm convention is

$\beta=1$7

For Turyn polynomials, G\"unther and Schmidt established the existence of a limiting profile $\beta=1$8 via

$\beta=1$9

They verified for $$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$0 that $$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$1 is minimized at $$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$2, and conjectured this for all $$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$3. The recent resolution is the theorem

$$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$4

which proves the conjecture in full (Bober et al., 7 Oct 2025).

The statement is asymptotic rather than finite-$$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$5. It identifies the minimizing shift of the limiting profile

$$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$6

but does not assert that for every sufficiently large prime $$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$7 the minimizing discrete shift must be exactly $$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$8 or any related finite statement. The contribution is the exact asymptotic minimizer, not an explicit closed-form value of $$S(\chi,\alpha,\beta,\theta)=\sum_{\alpha q<n<(\alpha+\beta)q}\chi(n)e(n\theta),$$9 (Bober et al., 7 Oct 2025).

The proof replaces the recursive combinatorics of G\"unther–Schmidt with a probabilistic representation derived from a quadratic random-process limit. For fixed $k\ge 2$0 and $k\ge 2$1, the random process $k\ge 2$2 converges in distribution to

$k\ge 2$3

where the $k\ge 2$4 are independent $k\ge 2$5-valued random variables. In the Turyn case $k\ge 2$6, this yields a probabilistic model for the shifted quadratic polynomial (Bober et al., 7 Oct 2025).

The limiting norm profile $k\ge 2$7 is then rewritten as an expectation over a random Fourier-type series, and a moment expansion shows that only pairings survive because the independence of the $k\ge 2$8 annihilates mixed terms unless indices occur in pairs. This reduces the minimization to the building block

$k\ge 2$9

Using an integral representation derived from the Lerch transcendent, the analysis shows that this is minimized at $L_{2k}$0, hence at $L_{2k}$1, which propagates back to the full theorem (Bober et al., 7 Oct 2025).

The quarter-shift theorem is therefore not merely a computational observation. It is an asymptotic extremal result valid for every even norm $L_{2k}$2 with $L_{2k}$3, and it supplies a conceptual explanation for the recurrent appearance of the quarter-shift in Turyn-polynomial flatness phenomena.

4. Flatness, Mahler measure, and record behavior near $L_{2k}$4

The Mahler-measure viewpoint places Turyn polynomials inside Mahler’s problem for Littlewood polynomials. For a polynomial

$L_{2k}$5

the Mahler measure is defined by

$L_{2k}$6

equivalently

$L_{2k}$7

For unimodular polynomials of degree $L_{2k}$8, Parseval gives $L_{2k}$9, so the natural normalized quantity is $\alpha=\tfrac14$0 (Mossinghoff, 2024).

Recent work analyzes Turyn polynomials through the random process

$\alpha=\tfrac14$1

where $\alpha=\tfrac14$2 are i.i.d. Rademacher random variables. For every $\alpha=\tfrac14$3 and every $\alpha=\tfrac14$4,

$\alpha=\tfrac14$5

where

$\alpha=\tfrac14$6

and for Mahler measure,

$\alpha=\tfrac14$7

where

$\alpha=\tfrac14$8

(Mossinghoff, 2024).

The numerical study reports that the data suggest $\alpha=\tfrac14$9 is increasing on $0.95$0, and for the quarter-shift it gives

$0.95$1

This is presented as a numerical estimate rather than a proved closed-form constant. The companion Littlewood polynomials $0.95$2 have the same asymptotic normalized Mahler measure, so the one-coefficient correction does not disturb the asymptotic value and yields a genuine $0.95$3-coefficient family with normalized Mahler measure exceeding $0.95$4 (Mossinghoff, 2024).

The asymptotic $0.95$5-norm minimization theorem provides an explanatory mechanism for this phenomenon. The quarter-shift Turyn polynomials are described as tending to be “somewhat flat,” and this flatness is presented as a natural reason to expect large Mahler measure. A plausible implication is that the Mahler-measure extremality and the even-norm extremality are manifestations of the same underlying distributional structure, though the Mahler result requires separate logarithmic analysis not used in the $0.95$6-proof (Bober et al., 7 Oct 2025).

5. Autocorrelation, Barker sequences, and Turyn-type sequence constructions

Turyn’s name also appears prominently in low-autocorrelation sequence theory, which supplies an adjacent but distinct polynomial context. A binary sequence $0.95$7 has aperiodic autocorrelation

$0.95$8

and a Barker sequence is one for which

$0.95$9

Encoding the sequence by the polynomial

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$00

the autocorrelations appear in

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$01

This makes Barker-sequence theory directly relevant to Turyn-style polynomial questions, even though the resulting objects are not the shifted quadratic-character Turyn polynomials of the analytic number-theory literature (Schmidt et al., 2015).

The odd-length case of Turyn’s conjecture is completely settled: if there exists a Barker sequence of odd length $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$02, then $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$03. The proof in the modern treatment proceeds via run structure, parity, two Turyn–Storer identities, and an autocorrelation-difference contradiction, rather than the original more intricate induction (Schmidt et al., 2015).

A broader flatness-based argument for normalized $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$04-coefficient Littlewood polynomials asserts that there are no square $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$05-flat sequences

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$06

and derives as a consequence that there are only finitely many Barker sequences. In that framework, square $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$07-flatness is equivalent to $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$08, so long Barker-type perfection would force impossible asymptotic flatness (Abdalaoui, 2016).

Another adjacent construction is the theory of Turyn-type sequences $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$09, quadruples $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$10 of $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$11-sequences whose associated polynomials satisfy the norm identity

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$12

This polynomial formulation is explicit, and the paper develops an equivalence relation, a canonical form, enumeration up to $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$13, and the first example of $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$14. These objects are not the same as shifted Legendre-symbol Turyn polynomials, but they belong to the same autocorrelation-centered combinatorial ecosystem and share the Littlewood-polynomial perspective (Best et al., 2012).

6. Terminological distinctions and conceptual synthesis

The modern literature supports a precise but layered usage of the term “Turyn polynomials.” In analytic number theory, the term refers to shifted Fekete polynomials or, equivalently, shifted quadratic-character polynomials

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$15

with normalized shift $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$16, or to the equivalent form

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$17

This is the sense in which the quarter-shift theorem and the Mahler-measure record are formulated (Mossinghoff, 2024).

In combinatorial design theory and autocorrelation theory, however, Turyn’s name also labels other constructions: Turyn-type sequences, Turyn–Golay conjectures, and Turyn–Storer identities. These are polynomially encoded through autocorrelation or Laurent-norm identities, but they are structurally distinct from the shifted Legendre-symbol family (Best et al., 2012). A plausible implication is that the shared nomenclature reflects a common focus on $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$18-coefficients, flatness, autocorrelation, and extremal behavior rather than a single unified polynomial class.

A further distinction is necessary between Turyn and Turán. Turán expressions, such as

$$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$19

belong to a different literature on weak Hurwitz stability, Bell polynomials, Chebyshev polynomials, and related zero-location problems. They are not Turyn polynomials, and the similarity of names is merely orthographic (Chasse et al., 2014).

Taken together, these strands show that Turyn polynomials occupy a central place at the interface of quadratic character sums, Littlewood polynomials, flatness theory, and autocorrelation extremals. Their most definitive modern property is now asymptotic: for every integer $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$20, the quarter-shift $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$21 uniquely minimizes the limiting normalized $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$22-norm profile $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$23 (Bober et al., 7 Oct 2025). Their most prominent current application is to Mahler’s problem, where the same quarter-shift regime produces a genuine Littlewood family with asymptotic normalized Mahler measure about $$F_{p,a}(t)=\sum_{n\le p}\left(\frac{n+a}{p}\right)e(nt),$$24 after the standard one-coefficient correction (Mossinghoff, 2024). This identifies the quarter-shifted Turyn family as a distinguished extremal locus within the broader theory of quadratic-character Littlewood polynomials.