Improved lower bounds for the Mahler measure of the Fekete polynomials

TL;DR

This paper proves a new lower bound for the Mahler measure of Fekete polynomials, showing it grows at least proportionally to the square root of the prime, improving previous bounds.

Contribution

It establishes a stronger universal lower bound for the Mahler measure of Fekete polynomials, advancing understanding of their size for large primes.

Findings

Mahler measure of Fekete polynomials is at least c√p for some c > 1/2

Improves previous lower bounds for large primes

Zeros of Fekete polynomials on the unit circle are key to the analysis

Abstract

We show that there is an absolute constant $c > 1/2$ such that the Mahler measure of the Fekete polynomials $f_p$ of the form $$f_p(z) := \sum_{k=1}^{p-1}{\left( \frac kp \right)z^k}\,,$$ (where the coefficients are the usual Legendre symbols) is at least $c\sqrt{p}$ for all sufficiently large primes $p$. This improves the lower bound $\left(\frac 12 - \varepsilon\right)\sqrt{p}$ known before for the Mahler measure of the Fekete polynomials $f_p$ for all sufficiently large primes $p \geq c_{\varepsilon}$. Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.

Equations

$$f_p(z):=\sum _{k=1}^{p-1}\left(\frac{k}{p}\right)z^k,$$

$${\text{Cal}K}_n:=\left\{P_n:P_n(z)=\sum _{k=0}^n{a}_k{z}^k,a_k\in \mathbb{C},|a_k|=1\right\}.$$

$${\text{Cal}L}_n:=\left\{P_n:P_n(z)=\sum _{k=0}^n{a}_k{z}^k,a_k\in \{-1,1\}\right\}.$$

$${\int }_0^{2\pi }|P_n(e^{it}){|}^{2}dt=2\pi (n+1)$$

$$\underset{z\in \partial D}{\min }|P_n(z)|<\sqrt{n+1}<\underset{z\in \partial D}{\max }|P_n(z)|$$

$$M_0(P,[\alpha ,\beta ]):=\exp \left(\frac{1}{\beta -\alpha }{\int }_{\alpha }^{\beta }\log |P(e^{it})|dt\right).$$

$$M_0(P,[\alpha ,\beta ])=\underset{q\to 0+}{\lim }{M}_q(P,[\alpha ,\beta ]),$$

$$M_q(P,[\alpha ,\beta ]):={\left(\frac{1}{\beta -\alpha }{\int }_{\alpha }^{\beta }{\left|P(e^{it})\right|}^{q}dt\right)}^{1/q}.$$

$$M_0(P):=M_0(P,[0,2\pi ])=|c|\prod _{k=1}^n\max \{1,|z_k|\}$$

$$P(z)=c\prod _{k=1}^n(z-z_k),c,z_k\in \mathbb{C}.$$

$$\underset{n\to \infty }{\lim }\frac{1}{2^{n+1}}\sum _{f\in {\text{Cal}L}_n}\frac{(M_q(f,[0,2\pi ]){)}^q}{n^{q/2}}=\Gamma \left(1+\frac{q}{2}\right)$$

$$\underset{n\to \infty }{\lim }\frac{1}{2^{n+1}}\sum _{f\in {\text{Cal}L}_n}\frac{M_q(f,[0,2\pi ])}{n^{1/2}}={\left(\Gamma \left(1+\frac{q}{2}\right)\right)}^{1/q}$$

$$\underset{n\to \infty }{\lim }\frac{1}{2^{n+1}}\sum _{f\in {\text{Cal}L}_n}\log \left(\frac{M_0(\hat{f},[0,2\pi ])}{n^{1/2}}\right)=-\gamma /2,$$

$$\underset{n\to \infty }{\lim }\frac{1}{2^{n+1}}\sum _{f\in {\text{Cal}L}_n}\frac{M_0(f,[0,2\pi ])}{n^{1/2}}=e^{-\gamma /2},$$

$$\gamma :=\underset{n\to \infty }{\lim }\left(\sum _{k=1}^n\frac{1}{k}-\log n\right)=0.577215\dots$$

$$f_p(z):=\sum _{k=1}^{p-1}\left(\frac{k}{p}\right)z^k,$$

$$\left(\frac{k}{p}\right)=\{\begin{array}{l}1,\text{if }{x}^2\equiv k(\text{mod}p)\text{for an}x\ne 0,\\ 0,\text{if }p\text{divides}k,\\ -1,\text{otherwise}\end{array}$$

$$c_1\sqrt{p}\log \log p\le \underset{z\in \partial D}{\max }|f_p(z)|\le c_2\sqrt{p}\log p.$$

$$M_0(f_p,[0,2\pi ])\ge \left(\frac{1}{2}-\epsilon \right)\sqrt{p}$$

$$M_0(f_p)\le M_2(f_p)=\sqrt{p-1}.$$

$$f_p(e^{2it})=e^{ipt}\sum _{k=0}^{(p-3)/2}{a}_k\cos ((2k+1)t),a_k\in \{-2,2\}.$$

$$\begin{array}{rl}{M}_0(f_p)\le & \frac{1}{2\pi }{\int }_0^{2\pi }|f_p(e^{iu})|du=\frac{1}{2\pi }{\int }_0^{\pi }|f_p(e^{2it})|2dt=\frac{1}{2\pi }{\int }_0^{2\pi }|f_p(e^{2it})|dt\\ \le & (1-{\epsilon }_0)\sqrt{p-1}\end{array}$$

$$i{f}_p(e^{2it})=e^{ipt}\sum _{k=0}^{(p-3)/2}{a}_k\sin ((2k+1)t),a_k\in \{-2,2\}.$$

$$\begin{array}{rl}{M}_0(f_p)& \le \frac{1}{2\pi }{\int }_0^{2\pi }|i{f}_p(e^{iu})|du=\frac{1}{2\pi }{\int }_0^{\pi }|i{f}_p(e^{2it})|2dt=\frac{1}{2\pi }{\int }_0^{2\pi }|i{f}_p(e^{2it})|dt\\ & \le (1-{\epsilon }_0)\sqrt{p-1}\end{array}$$

$$M_0(f_n)\ge (1-\epsilon )\sqrt{n}.$$

$$M_0(f_p,[\alpha ,\beta ])\ge c_1{p}^{1/2}$$

$$M_q(f_p,[\alpha ,\beta ])\le c_2(q,\epsilon )p^{1/2},$$

$$c_1{p}^{1/2}\le M_q(f_p,[\alpha ,\beta ])\le c_2(q,\epsilon )p^{1/2}$$

$$c_1\sqrt{n+1}\le |P_n(z)|\le c_2\sqrt{n+1},z\in \partial D,$$

$$\begin{array}{rl}{P}_0(z)& :=1,Q_0(z):=1,\\ P_{n+1}(z)& :=P_n(z)+z^{2^n}{Q}_n(z),\\ Q_{n+1}(z)& :=P_n(z)-z^{2^n}{Q}_n(z),n=0,1,2,\dots .\end{array}$$

Full text

Improved lower bounds for the Mahler measure of the Fekete polynomials

Tamás Erdélyi

Department of Mathematics, Texas A&M University, College Station, Texas 77843, College Station, Texas 77843 (T. Erdélyi)

(February 17, 2017 )

We show that there is an absolute constant $c>1/2$ such that the Mahler measure of the Fekete polynomials $f_{p}$ of the form

[TABLE]

(where the coefficients are the usual Legendre symbols) is at least $c\sqrt{p}$ for all sufficiently large primes $p$. This improves the lower bound $\left(\frac{1}{2}-\varepsilon\right)\sqrt{p}$ known before for the Mahler measure of the Fekete polynomials $f_{p}$ for all sufficiently large primes $p\geq c_{\varepsilon}$. Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.

polynomials, restricted coefficients, number of zeros on the unit circle, Legendre symbols, Fekete polynomials, Mahler measure

††support: 2010 Mathematics Subject Classifications. 11C08, 41A17, 26C10, 30C15

1. Introduction and Notation

Let $D$ be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by $\partial D$. Let

[TABLE]

The class ${\Cal{K}}_{n}$ is often called the collection of all (complex) unimodular polynomials of degree $n$. Let

[TABLE]

The class ${\Cal{L}}_{n}$ is often called the collection of all Littlewood polynomials of degree $n$. By Parseval’s formula,

[TABLE]

for all $P_{n}\in{\Cal{K}}_{n}$. Therefore

[TABLE]

for all $P_{n}\in{\Cal{K}}_{n}$ and $n\geq 1$. An old problem (or rather an old theme) is the following.

Let $\alpha<\beta$ be real numbers. The Mahler measure $M_{0}(P,[\alpha,\beta])$ is defined for bounded measurable functions $P(e^{it})$ defined on $[\alpha,\beta]$ as

[TABLE]

It is well known that

[TABLE]

where, for $q>0$,

[TABLE]

It is a simple consequence of the Jensen formula that

[TABLE]

for every polynomial of the form

[TABLE]

P. Borwein and Lockhart [B-01] investigated the asymptotic behavior of the mean value of normalized $L_{q}$ norms of Littlewood polynomials for arbitrary $q>0$. Using the Lindeberg Central Limit Theorem and dominated convergence, they proved that

[TABLE]

for every $q>0$. In [C-15] we proved that

[TABLE]

for every $q>0$. We also proved analogous results for the Mahler measure. Namely, using the notation $\widehat{f}(z):=\max\{|f(z)|,n^{-1}\}$, we have

[TABLE]

and

[TABLE]

where

[TABLE]

is the Euler constant and $e^{-\gamma/2}=0.749306\ldots$. These are analogues of the results proved earlier by Choi and Mossinghoff [C-11] for polynomials in ${\Cal{K}}_{n}$.

Finding polynomials with suitably restricted coefficients and maximal Mahler measure has interested many authors. Beller and Newman [B-73] constructed unimodular polynomials of degree $n$ whose Mahler measure is at least $\sqrt{n}-c/\log n$. For a prime $p$ the $p$-th Fekete polynomial is defined as

[TABLE]

where

[TABLE]

is the usual Legendre symbol. Note that $g_{p}(z):=f_{p}(z)/z$ is a Littlewood polynomial, and has the same Mahler measure as $f_{p}$.

Montgomery [M-80] proved the following fundamental result.

Theorem 1.1

There are absolute constants $c_{1}>0$ and $c_{2}>0$ such that

[TABLE]

In [E-07] we proved the following result.

Theorem 1.2

For every $\varepsilon>0$ there is a constant $c_{\varepsilon}$ such that

[TABLE]

for all primes $p\geq c_{\varepsilon}$.

From Jensen’s inequality,

[TABLE]

However, as it was observed in [E-07], $\frac{1}{2}-\varepsilon$ in Theorem 1.2 cannot be replaced by $1-\varepsilon$. Indeed, if $p\geq 3$ is a prime and $p=4m+1$, then $f_{p}$ is self-reciprocal, that is, $z^{p}f_{p}(1/z)=f_{p}(z)$, and hence

[TABLE]

Therefore a result of Littlewood [L-66] implies that

[TABLE]

with some absolute constant $\varepsilon_{0}>0$. If $p\geq 3$ is a prime and $p=4m+3$, then $f_{p}$ is anti-self-reciprocal, that is, $z^{p}f_{p}(1/z)=-f_{p}(z)$, and hence

[TABLE]

Therefore a result of Littlewood [L-66] implies that

[TABLE]

with some absolute constant $\varepsilon_{0}>0$.

It is an interesting open question whether there is a sequence of Littlewood polynomials $(f_{n})$ such that for an arbitrary $\varepsilon>0$, and $n$ large enough,

[TABLE]

In [E-11] Theorem 1.2 was extended to subarcs of the unit circle.

Theorem 1.3

There exists an absolute constant $c_{1}>0$ such that

[TABLE]

for all primes $p$ and for all $\alpha,\beta\in{\mathbb{R}}$ such that $(\log p)^{3/2}p^{-1/2}\leq\beta-\alpha\leq 2\pi$.

In [E-12] we gave an upper bound for the average value of $|f_{p}(z)|^{q}$ over any subarc $I$ of the unit circle, valid for all sufficiently large primes $p$ and all exponents $q>0$.

Theorem 1.4

There exists a constant $c_{2}(q,\varepsilon)$ depending only on $q>0$ and $\varepsilon>0$ such that

[TABLE]

for all primes $p$ and for all $\alpha,\beta\in{\mathbb{R}}$ such that $2p^{-1/2+\varepsilon}\leq\beta-\alpha\leq 2\pi$.

We remark that a combination of Theorems 1.3 and 1.4 shows that there is an absolute constant $c_{1}>0$ and a constant $c_{2}(q,\varepsilon)>0$ depending only on $q>0$ and $\varepsilon>0$ such that

[TABLE]

for all primes $p$ and for all $\alpha,\beta\in{\mathbb{R}}$ such that $(\log p)^{3/2}p^{-1/2}\leq 2p^{-1/2+\varepsilon}\leq\beta-\alpha\leq 2\pi$.

The $L_{q}$ norm of polynomials related to Fekete polynomials were studied in several recent papers. See [B-01b], [B-02], [B-04], [G-16], [J-13a], and [J-13b], for example. An interesting extremal property of the Fekete polynomials is proved in [B-01c].

Fekete might have been the first one to study analytic properties of the Fekete polynomials. He had an idea of proving non-existence of Siegel zeros (that is, real zeros “especially close to $1$”) of Dirichlet $L$-functions from the positivity of Fekete polynomials on the interval $(0,1)$, where the positivity of Fekete polynomials is often referred to as the Fekete Hypothesis. There were many mathematicians trying to understand the zeros of Fekete polynomials including Fekete and Pólya [F-12], Pólya [P-19], Chowla [C-35], Heilbronn [H-37], Montgomery [M-80], Baker and Montgomery [B-90], and Jung and Shen [J-16].

Baker and Montgomery [B-90] proved that $f_{p}$ has a large number of zeros in $(0,1)$ for almost all primes $p$, that is, the number of zeros of $f_{p}$ in $(0,1)$ tends to $\infty$ as $p$ tends to $\infty$, and it seems likely that there are, in fact, about $\log\log p$ such zeros.

Conrey, Granville, Poonen, and Soundararajan [C-00] showed that $f_{p}$ has asymptotically $\kappa p$ zeros on the unit circle, where $0.500668<\kappa<0.500813$.

An interesting recent paper [B-17] studies power series approximations to Fekete polynomials.

It is conjectured, see [B-02] for instance, that there are sequences of flat Littlewood polynomials $P_{n}\in{\Cal{L}}_{n}$ satisfying

[TABLE]

with absolute constants $c_{1}>0$ and $c_{2}>0$. However, the lower bound part of this conjecture, by itself, seems hard, and no sequence is known that satisfies just the lower bound. A sequence of Littlewood polynomials satisfying just the upper bound is given by the Rudin-Shapiro polynomials. They appear in Harold Shapiro’s 1951 thesis [S-51] at MIT and are sometimes called just Shapiro polynomials. They also arise independently in a paper by Golay (1951). They are remarkably simple to construct and are a rich source of counterexamples to possible conjectures. The Rudin-Shapiro polynomials are defined recursively as follows:

[TABLE]

Note that both $P_{n}$ and $Q_{n}$ are polynomials of degree $N-1$ with $N:=2^{n}$ having each of their coefficients in $\{-1,1\}$. In [E-16] we showed that the Mahler measure and the maximum norm of the Rudin-Shapiro polynomials on the unit circle of the complex plane have the same size.

Theorem 1.5

Let $P_{n}$ and $Q_{n}$ be the $n$-th Rudin-Shapiro polynomials defined in Section 1. There is an absolute constant $c_{1}>0$ such that

[TABLE]

where

[TABLE]

2. New Result

In this paper we improve the factor $\left(\frac{1}{2}-\varepsilon\right)$ in Theorem 1.1 to an absolute constant $c>1/2$. Namely we prove the following.

Theorem 2.1

There is an absolute constant $c>1/2$ such that

[TABLE]

for all sufficiently large primes.

3. Lemmas

To prove the theorem we need a few lemmas. For a natural number $p$ let

[TABLE]

be the first $p$-th root of unity. Our first lemma formulates an characteristic property of the Fekete polynomials. A simple proof is given in [B-02, pp. 37-38].

Lemma 3.1 (Gauss)

We have

[TABLE]

and $f_{p}(1)=0$.

Lemma 3.2

We have

[TABLE]

for all polynomials $Q$ of degree at most $p$ with complex coefficients.

Lemma 3.3

Let $0<\eta\leq\pi/2$ be fixed. Suppose a polynomial $Q$ of degree at most $p$ with complex coefficients has at least $k$ zeros

[TABLE]

such that

[TABLE]

We have

[TABLE]

In the proof of Theorem 2.1 we need one of the following two results. For proofs see [B-97a] and [B97-b], respectively.

Lemma 3.4

There is an absolute constant $c>0$ such that every $Q\in{\Cal{K}}_{n}$ has at most $c\sqrt{n}$ real zeros.

Lemma 3.5

There is an absolute constant $c>0$ such that every $Q\in{\Cal{L}}_{n}$ has at most $\displaystyle{\frac{c\log^{2}n}{\log\log n}}$ zeros at $1$.

The large sieve of number theory [M-78] asserts the following.

Lemma 3.6

If

[TABLE]

is a trigonometric polynomial of degree at most $n$,

[TABLE]

and

[TABLE]

then

[TABLE]

It turns out to be fairly easy to show that at least half of the zeros of $f_{p}$ are on the unit circle $\partial D$. First note that

[TABLE]

Observe also that

[TABLE]

Define $H_{p}(t):=F_{p}(e^{2i\pi t})$ if $p\equiv 1$ mod $\,4$, and $H_{p}(t):=-iF_{p}(e^{2i\pi t})$ if $p\equiv 3$ mod $\,4$. By (3.1) we see that $H_{p}(t)$ is a periodic, continuous, real-valued function when $t$ is real.

Lemma 3.7

Let $p$ be a prime. There are at least $(p-3)/2$ values of $k\in\{0,1,\ldots,p-1\}$ for which $H_{p}$ has a zero between $k/p$ and $(k+1)/p$.

Our next lemma is Theorem 4 in [C-00]. For a proof of Lemma 3.8 below see Section 6 in [C-00].

Lemma 3.8

Let $p$ be a prime. For every fixed real number $\delta$

[TABLE]

as $p\rightarrow\infty$, where

[TABLE]

Moreover $c_{-\delta}=1-c_{\delta}$ for all $\delta>0$.

Lemma 3.9

For every $\varepsilon>0$ there is a $\delta>0$ such that

[TABLE]

for all sufficiently large primes $p\geq N_{\varepsilon}$.

Lemma 3.10

Let $\gamma>0$ be a real number. Let the subarcs $I_{k}$ of the unit circle $\partial D$ be defined by

[TABLE]

We have

[TABLE]

for all primes $p\geq 3$.

Lemma 3.11

Given $\eta>0$ let the subarcs $I_{k,\eta}$ of the unit circle $\partial D$ be defined by

[TABLE]

For every $\varepsilon>0$ there is an $\eta>0$ such that

[TABLE]

for all sufficiently large primes $p\geq N_{\varepsilon}$.

4. Proofs of the Lemmas

Demonstration Proof of Lemma 3.2

Let

[TABLE]

with some $m\leq p$. Without loss of generality we may assume that $c=1$. Note that

[TABLE]

while

[TABLE]

Multiplying these inequalities for $j=1,2,\ldots,m$, we obtain

[TABLE]

∎

Demonstration Proof of Lemma 3.3

Let

[TABLE]

with some $m\leq p$, where $a_{j}=b_{j}$, $j=1,2,\ldots,k$. Without loss of generality we may assume that $c=1$. Note that

[TABLE]

and

[TABLE]

Multiplying these inequalities for $j=1,2,\ldots,m$, we obtain

[TABLE]

∎

Demonstration Proof of Lemma 3.7

By Lemma 3.1 If $\zeta_{p}=e^{2i\pi/p}$ then, for all $k$ not divisible by $p$ we have $|f_{p}(\zeta_{p}^{k})|=\sqrt{p}$, and hence $|F_{p}(\zeta_{p}^{k})|=\sqrt{p}$. Moreover

[TABLE]

Therefore if $\displaystyle{\left(\frac{k}{p}\right)=\left(\frac{k+1}{p}\right)}$, then $\displaystyle{H_{p}\left(\frac{k}{p}\right)}$ and $\displaystyle{H_{p}\left(\frac{k+1}{p}\right)}$ have different signs. Since $H_{p}(t)$ is real-valued and continuous on the real line, it must have a zero $k/p$ and $(k+1)/p$ by the Intermediate Value Theorem. However, by Lemma 2 in [C-00] we have

[TABLE]

and hence the values of $k\in\{0,1,\ldots,p-1\}$ for which $H_{p}$ has a zero between $k/p$ and $(k+1)/p$ is at least $(p-3)/2$. ∎

Demonstration Proof of Lemma 3.9

Note that

[TABLE]

converges for every fixed $\delta>0$, and

[TABLE]

Indeed, there is an absolute constant $c_{1}>0$ such that

[TABLE]

as

[TABLE]

Also,

[TABLE]

Therefore

[TABLE]

where

[TABLE]

and

[TABLE]

So by choosing $\delta>0$ so that

[TABLE]

the lemma follows from Lemma 3.8. ∎

Demonstration Proof of Lemma 3.10

Suppose there are $0\leq k_{1}<k_{2}<\cdots<k_{m}\leq p-1$ such that

[TABLE]

Then

[TABLE]

and

[TABLE]

Hence by the large sieve inequality formulated in Lemma 3.6 and the Parseval formula appliied to $g_{p}(z):=z^{(3-p)/2}f_{p}^{\prime}(z)$ we get

[TABLE]

∎

Demonstration Proof of Lemma 3.11

Let $\varepsilon>0$. By Lemma 3.9 there is a $\delta>0$ depending only on $\varepsilon>0$ such that

[TABLE]

for all sufficiently large primes $p\geq N_{\varepsilon}$. Let $\gamma:=\varepsilon^{-1/2}$. By Lemma 3.10 we have

[TABLE]

Now let

[TABLE]

By (4.1) and (4.2) we obtain

[TABLE]

Let $0<\eta<\min\{\delta/\gamma,\pi/2\}$. Observe that $k\in A_{p,\delta,\gamma}$ implies that $f_{p}$ does not vanish in $I_{k,\eta}$. Indeed, $z:=e^{it}\in I_{k,\eta}$ implies

[TABLE]

and hence

[TABLE]

for all sufficiently large primes $p\geq N_{\varepsilon}$, and the lemma follows from (4.3). ∎

Proof of Theorem 2.1

Now we are ready to prove the theorem.

Demonstration Proof of Theorem 2.1

As in Lemma 3.11 let the subarcs $I_{k,\eta}$ of the unit circle ${\partial D}$ be defined by

[TABLE]

It follows from Lemma 3.11 that for $\varepsilon:=1/8$ there is an $\eta>0$ such that

[TABLE]

for all sufficiently large primes $p$. Combining this with Lemma 3.7 we have that

[TABLE]

for all sufficiently large primes $p$. Hence the assumptions of Lemma 3.3 are satisfied with $Q:=f_{p}$ and $k\geq p/4$ for all sufficiently large primes. Suppose that $1$ is a zero of $f_{p}$ with multiplicity $m=m(p)$. By either Lemma 3.4 or Lemma 3.5 we have $m=O(p^{1/2})$. Let $g_{p}(z):=f_{p}(z)/h_{m}(z)$ with $h_{m}(z):=(z-1)^{m}$. Note that $|g_{p}(1)|$ is a nonzero integer, hence $|g_{p}(1)|\geq 1$. Also, $h_{m}$ is monic and has all its zeros on the unit circle, hence $M_{0}(h_{m})=1$. Combining these with the multiplicative property of the Mahler measure, Lemma 3.3 applied to $g_{p}$ with $k\geq p/4$, Lemma 3.1, and the fact that $m=O(p^{1/2})$ implies that

[TABLE]

we conclude that there are absolute constants

[TABLE]

such that

[TABLE]

for all sufficiently large primes $p$. ∎

5. Acknowledgment

The author thanks Stephen Choi for his careful reading of the paper and for his comments.