Distribution of Small Values of Bohr Almost Periodic Functions with Bounded Spectrum

TL;DR

This paper establishes bounds on the measure of small-value sets of Bohr almost periodic functions with bounded spectrum, linking these bounds to the Mahler measure and the Riemann Hypothesis.

Contribution

It proves a universal measure estimate for small values of such functions and relates the Mahler measure of their lifts to deep number theory conjectures.

Findings

Mean measure of small-value sets is bounded by a power of u.

For trigonometric polynomials, the bound depends only on the number of frequencies and largest coefficient.

Positivity of Mahler measure is established, with implications for the Riemann Hypothesis.

Abstract

If $f$ is a nonzero Bohr almost periodic function on $\mathbb R$ with a bounded spectrum we prove there exist $C_f > 0$ and integer $n > 0$ such that for every $u > 0$ the mean measure of the set $\{\, x \, : \, |f(x)| < u \, \}$ is less than $C_f\, u^{1/n}.$ For trigonometric polynomials with $\leq n + 1$ frequencies we show that $C_f$ can be chosen to depend only on $n$ and the modulus of the largest coefficient of $f.$ We show this bound implies that the Mahler measure $M(h),$ of the lift $h$ of $f$ to a compactification $G$ of $\mathbb R,$ is positive and discuss the relationship of Mahler measure to the Riemann Hypothesis.

Equations

$$J_f(u):=m\left(\{x\in \mathbb{R}:|f(x)|<u\}\right)$$

$$J_f(u)\le C_f{u}^{\frac{1}{n}},u>0.$$

$$J_f(u)\le C_n{H}_f^{-\frac{1}{n}}{u}^{\frac{1}{n}},u>0.$$

$${\Xi }_{f,\omega ,k,u}(v):=m\{x\in \mathbb{R}:|f(x)|<u,|({\chi }_{\omega }f{)}^{(j)}(x)|<v^j,j=1,...,k\},$$

$$K_f(u):=\underset{\omega \in \mathbb{R}}{\inf }\underset{k\in \mathbb{N}}{\inf }\underset{v>0}{\inf }\left[3\sqrt{2}{\pi }^{-1}b(f)k{v}^{-1}{u}^{\frac{1}{k}}+{\Xi }_{f,\omega ,k,u}(v)\right].$$

$$\underset{k\to \infty }{\mathrm{limsup}}|f^{(k)}(x){|}^{\frac{1}{k}}=\frac{b(f)}{2}$$

$$f(x)=\sum _{j=1}^{n+1}{a}_j{e}^{i{\omega }_{j}x},0={\omega }_1<\cdots <{\omega }_{n+1}=1,H_f=\max \{|a_j|:j=2,...,n+1\}.$$

$$J_{f^{(1)}}(v)\le C_{n-1}{H}_f^{\frac{1}{n-1}}{v}^{\frac{1}{n-1}},$$

$${\Xi }_{f,0,1,u}(v)\le C_{n-1}{H}_f^{\frac{1}{n-1}}{v}^{\frac{1}{n-1}}.$$

$$J_f(u)\le \underset{v>0}{\inf }\left[3\sqrt{2}{\pi }^{-1}{v}^{-1}u+C_{n-1}{H}_f^{\frac{1}{n-1}}{v}^{\frac{1}{n-1}}\right]=C_n{H}_f^{\frac{1}{n}}{u}^{\frac{1}{n}}$$

$$where{C}_n:=C_{n-1}^{1-\frac{1}{n}}[3\sqrt{2}{\pi }^{-1}(n-1){]}^{\frac{1}{n}}n(n-1{)}^{-1}.$$

$$b-a\le 2\sqrt{2}\frac{\max |\varphi |([a,b])}{\min |{\varphi }^{\prime }|([a,b])}$$

$$(b-a)\min |{\varphi }^{\prime }|([a,b])\le {\int }_a^b|{\varphi }^{\prime }(y)|dy\le {\int }_a^b\left(|\Re {\varphi }^{\prime }(y)|+|\Im {\varphi }^{\prime }(y)|\right)dy.$$

$${\int }_a^b\left(|\Re {\varphi }^{\prime }(y)|+|\Im {\varphi }^{\prime }(y)|\right)dy=(c\Re \varphi (b)+d\Im \varphi (b))-(c\Re \varphi (a)+d\Im \varphi (a)).$$

$$S_{f,k,u,v}:=\{x\in \mathbb{R}:|f(u)|<u,\underset{j\in \{1,...,k\}}{\max }|f^{(j)}(x){|}^{\frac{1}{j}}\ge v\}.$$

$$It\; suffices\; to\; prove\; thats(f,k,u,v)\le 3\sqrt{2}{\pi }^{-1}b(f)k{v}^{-1}{u}^{\frac{1}{k}}.$$

$$\psi :=\prod _{j=0}^{k-1}(\Re f^{(j+1)})(\Im f^{(j+1)})(|f^{(j)}(x){|}^2-{\gamma }_j^2)(|f^{(j+1)}(x){|}^2-{\gamma }_{j+1}^2).$$

$$Lemma\text{ref}lem2implies\; thatlength(I)\le 2\sqrt{2}\frac{{\gamma }_k}{{\gamma }_{k+1}}=2\sqrt{2}{v}^{-1}{u}^{\frac{1}{k}},I\in \mathcal{I},$$

$$and\; (\text{ref}S)\; and\; Property\; 3\; implies\; that{S}_{f,k,u,v}\subset \bigcup _{I\in \mathcal{I}}I.$$

$$m(|{\log }^{-}\circ |f|{|}^p)\le {\int }_0^1|\log (u){|}^{p}d{C}_f{u}^{\frac{1}{n}}=C_f{n}^p\Gamma (p),$$

$${\int }_G|{\log }^{-}\circ |h{|}_j{|}^p=m(|{\log }^{-}\circ |f{|}_j{|}^p)\le m(|{\log }^{-}\circ |f|{|}^p).$$

$${\int }_G\log \circ |h|=\log \left|{\int }_{G}h\right|.$$

$$\log M^{+}({\Phi }_N){\ll }_{ϵ}{N}^{\frac{1}{2}+ϵ},ϵ>0.$$

$$\log M^{+}({\Phi }_N)=-{\int }_0^1\log (u)d{J}_{f_N}(u).$$

$$P_n(z):=1+z+\cdots +z^n;Q_n(z):={\left(\genfrac{}{}{0ex}{}{n}{[n/2]}\right)}^{-1}(1+z{)}^n$$

$$J_{p_n}(u)\le \frac{2}{\pi }{\sin }^{-1}\left(\min \{1,u\}\right)\le u\Rightarrow \log (M^{-}(P_n))>-1,$$

$$J_{q_n}(u)=\frac{2}{\pi }{\sin }^{-1}\left(\min \left\{1,\frac{1}{2}{\left(\genfrac{}{}{0ex}{}{n}{[n/2]}\right)}^{\frac{1}{n}}{u}^{\frac{1}{n}}\right\}\right)\ge \frac{2}{\pi }{u}^{\frac{1}{n}}\Rightarrow \log (M^{-}(Q_n))<-\frac{2n}{\pi }.$$

Full text

2010 Mathematics Subject Classification : 11K70, 30D15, 11R06, 47A68

1 Distribution of Small Values

$\mathbb{N}\mathrel{\mathop{:}}=\{1,2,...\},\mathbb{Z},\mathbb{R},\mathbb{C},\mathbb{T}\mathrel{\mathop{:}}=\{\,z\in\mathbb{C}\,:\,|z|=1\ \}$ are the natural, integer, real, complex and circle group numbers, $\textrm{C}_{b}(\mathbb{R})$ is the $\textrm{C}^{*}$-algebra of bounded continuous functions and $\chi_{\omega}:\mathbb{R}\rightarrow\mathbb{T},\ \omega\in\mathbb{R}$ the homomorphisms $\chi_{\,\omega}(x)\mathrel{\mathop{:}}=e^{i\omega x},\ \omega\in\mathbb{R}.$ A finite sum $f=\sum_{\,\omega}a_{\,\omega}\,\chi_{\omega}$ with distinct $\omega$ is called a trigonometric polynomial with height $H_{f}\mathrel{\mathop{:}}=\max_{\,\omega}|a_{\,\omega}|$ and they comprise the algebra $\textrm{T}(\mathbb{R})$ of trigonometric polynomials. Bohr [9] defined the $\textrm{C}^{*}$-algebra $\textrm{U}(\mathbb{R})$ of uniformly almost periodic functions to be the closure of $\textrm{T}(\mathbb{R})$ in $\textrm{C}_{b}(\mathbb{R})$ and proved that their means $m(f)\,\mathrel{\mathop{:}}=\lim_{L\rightarrow\infty}(2L)^{-1}\int_{-L}^{L}f(t)dt$ exist. The Fourier transform $\widehat{f}\,:\,\mathbb{R}\rightarrow\mathbb{C}$ of $f\in\textrm{U}(\mathbb{R})$ is $\widehat{f}(\omega)\mathrel{\mathop{:}}=m(f\,\overline{\chi}_{\omega}\,)$ and its spectrum $\Omega(f)\mathrel{\mathop{:}}=\hbox{ support }\widehat{f}.$ If $f$ is nonzero then $\Omega(f)$ is nonempty and countable and we say $f$ has bounded spectrum if its bandwidth $b(f)\in[0,\infty],$ defined by $b(f)\mathrel{\mathop{:}}=\sup\Omega(f)-\inf\Omega(f),$ is finite. We observe that if $S\subseteq\mathbb{R}$ is defined by a finite number of inequalities involving functions in $\textrm{U}(\mathbb{R})$ then $m(S)\,\mathrel{\mathop{:}}=\lim_{L\rightarrow\infty}(2L)^{-1}\hbox{measure }[-L,L]\cap S$ exists and define $J_{f}:(0,\infty)\rightarrow[0,1]$ by

[TABLE]

Theorem 1

If $f\in{\textrm{U}}(\mathbb{R})$ is nonzero and has a bounded spectrum then there exist $C_{f}>0$ and $n\in\mathbb{N}$ such that:

[TABLE]

There exists a sequence $C_{n}$ such that if $f\in\textrm{T}(\mathbb{R})$ has $n+1$ frequencies then

[TABLE]

Proof For $f\in\textrm{U}(\mathbb{R}),\omega\in\mathbb{R},k\in\mathbb{N},u>0$ define $\Xi_{f,\omega,k,u}\,,K_{f}\,:\,(0,\infty)\rightarrow[0,1]$ by

[TABLE]

[TABLE]

We first prove Theorem 1 assuming the following result which we prove latter.

Lemma 1

Every nonzero $f\in\textrm{U}(\mathbb{R})$ with bounded spectrum satisfies $J_{f}\leq K_{f}.$

We observe that for every $\omega\in\mathbb{R}$ and every $a\in\mathbb{R}\backslash\{0\},$ if $h(x)=\chi_{\,\omega}(x)\,f(ax)$ then $J_{h}=J_{f}$ and $K_{h}=K_{f}.$ Without loss of generality we can assume that $\Omega(f)\subset[-\frac{b(f)}{2},\frac{b(f)}{2}].$ If $b(f)=0$ then $f=c$ and $J_{f}(u)\leq|c|^{-1}u.$ If $\Omega(f)>0$ then Bohr [10] proved that $f$ extends to an entire function $F$ of exponential type $\frac{b(f)}{2},$ and Boas [6], ([7], p. 11, Equation 2.2.12) proved that

[TABLE]

uniformly in $x.$ Therefore for any $v_{0}>\frac{b(f)}{2}$ there exists $k\in\mathbb{N}$ such that $\Xi_{f,0,k,u}(v_{0})=0$ so Lemma 1 implies $J_{f}$ satisfies (2) with $C_{f}=3\,{\sqrt{2}}\,\pi^{-1}\,b(f)\,k\,v_{0}^{-1}$ and $n=k.$ This proves the first assertion. To prove the second we assume, without loss of generality, that $b(f)=1,$ $\Omega(f)\subset[0,1]$ and

[TABLE]

Define $C_{1}\mathrel{\mathop{:}}=\frac{1}{2}.$ If $n=1$ and $f$ has $n+1=2$ terms and $f=a_{0}+a_{1}\chi_{1}$ with $|a_{1}|=H_{f}$ and $h=H_{f}(1-\chi_{1}),$ then $J_{f}(u)\leq J_{h}(u)=(2/\pi)\sin^{-1}(u/(2H_{f}))\leq C_{1}H_{f}^{-1}u$ therefore (3) holds for $n=1.$ For $n\geq 2$ we assume by induction that (3) holds for $n-1$ and therefore, since $f^{(1)}$ has $n$ terms and $H_{f^{(1)}}=H_{f},$ it follows that for all $v>0,$

[TABLE]

[TABLE]

Therefore Lemma 1 with $\omega=0,\,b(f)=k=1$ gives

[TABLE]

[TABLE]

Remark 1

Computation of 200 million terms shows that $n^{-1}C_{n}\rightarrow 0.900316322$

Conjecture 1

In (3) $C_{n}$ can be replaced by a bounded sequence.

Lemma 2

If $\phi:[a,b]\rightarrow\mathbb{C}$ is differentiable and $\phi^{\prime}([a,b])$ is contained in a quadrant then

[TABLE]

Proof of Lemma 2 We first proved this result in ([18], Lemma 1) where we used it to give a proof, of a conjecture of Boyd [11] about monic polynomials related to Lehmer’s conjecture [21], which was reviewed in ([13], Section 3.5) and extended to monic trigonometric polynomials in ([19], Lemma 2). The triangle inequality $|\phi^{\prime}|\leq|\Re\,\phi^{\prime}|+|\Im\,\phi^{\prime}|$ gives

[TABLE]

Since $\phi^{\prime}([a,b])$ is contained in a quadrant of $\mathbb{C}$ there exist $c,d\in\{1,-1\}$ such that $|\Re\,\phi^{\prime}(y)|=c\,\Re\,\phi^{\prime}(y)$ and $|\Im\,\phi^{\prime}(y)|=d\,\Im\,\phi^{\prime}(y)$ for all $y\in[a,b].$ Therefore

[TABLE]

The result follows since the right side is bounded above by $2{\sqrt{2}}\,\max|\phi|([a,b]).$

Proof of Lemma 1 Assume that $f\in\textrm{U}(\mathbb{R})$ is nonzero. We may assume without loss of generality that $\Omega(f)\subset[-\frac{b(f)}{2},\frac{b(f)}{2}].$ For $k\in\mathbb{N},u>0,v>0$ we define the set

[TABLE]

We observe that the set of functions in $\textrm{U}(\mathbb{R})$ whose spectrums are in $[-\frac{b(f)}{2},\frac{b(f)}{2}]$ is closed under differentiation, and define $s(f,k,u,v)\mathrel{\mathop{:}}=m(S_{f,k,u,v}).$

[TABLE]

Define $\gamma_{j}\mathrel{\mathop{:}}=u^{\frac{k-j}{k}}\,v^{j},j\in\{0,...,k\},$ and $\mathcal{I}\mathrel{\mathop{:}}=$ set of closed intervals $I$ satisfying, for some $j\in\{0,1,...,k-1\},$ the following three properties:

$f^{(j+1)}(I)$ is a subset of a closed quadrant, 2. 2.

$\max|f^{(j)}|(I)\leq\gamma_{j}$ and $\min|f^{(j+1)}|(I)\geq\gamma_{j+1},$ 3. 3.

$I$ is maximum with respect to properties 1 and 2.

Define $\mathcal{E}\mathrel{\mathop{:}}=$ set of endpoints of intervals in $\mathcal{I}$, and

[TABLE]

[TABLE]

[TABLE]

Clearly $\psi=\Psi|_{\mathbb{R}}$ where $\Psi$ is the product of $6k$ entire functions each having bandwidth $b(f)$ so a theorem of Titchmarsh [25] implies that the density of real zeros of $\Psi$ is bounded above by $3\pi^{-1}\,b(f)\,k.$ Property 3 implies that all points in $\mathcal{E}$ are zeros of $\Psi$ so the upper density of intervals in $\mathcal{I}$ is bounded by $\frac{3}{2}\,\pi^{-1}\,b(f)\,k.$ Combining these facts gives $s(f,k,u,v)\leq(\frac{3}{2}\pi^{-1}\,b(f)\,k)\,(2\sqrt{2}v^{-1}u^{\frac{1}{k}})=3\sqrt{2}\pi^{-1}\,b(k)\,k\,v^{-1}u^{\frac{1}{k}}$ which proves (13) and concludes the proof of Lemma 1.

For $p\in[1,\infty)$ Besicovitch [4] proved that the completion $\textrm{B}^{p}(\mathbb{R})$ of $\textrm{U}(\mathbb{R})$ with norm $(m(|f|^{p}))^{\frac{1}{p}}$ is a subset of $L_{loc}^{p}(\mathbb{R}).$ For $x\geq 0$ we define $\log^{+}(x)\mathrel{\mathop{:}}=\log(\max\{1,x\})\in[0,\infty),$ $\log^{-}(x)\mathrel{\mathop{:}}=\log(\min\{1,x\})\in[-\infty,0],$ and $|x|_{j}\mathrel{\mathop{:}}=\max\{|x|,\frac{1}{j}\}$ for $j\in\mathbb{N}.$

Corollary 1

If $f\in\textrm{U}(\mathbb{R})$ satisfies (2), then $\log^{-}\circ\,|f|\in\textrm{B}^{p}(\mathbb{R}),$

[TABLE]

and $\log\circ|f|\in B^{p}(\mathbb{R}).$

Proof of Corollary 1 Since the means of the functions $\log^{-}\circ|f|_{j}\,|^{p}$ are nondecreasing and bounded by the right side of (17), the sequence $\log^{-}\circ|f|_{j}$ is a Cauchy sequence in $B^{p}(\mathbb{R})$ so it converges to a function $\eta\in B^{p}(\mathbb{R}).$ Therefore $\log^{-}\circ|f|=\eta$ since it is the pointwise limit of $\log^{-}\circ|f|_{j}$ and $\eta\in L_{loc}^{p}(\mathbb{R}).$ The last fact follows since $\log=\log^{+}+\log^{-}.$

2 Compactifications and Hardy Spaces

Definition 1

A compactification of $\mathbb{R}$ is a pair $(G,\theta)$ where $G$ is a compact abelian group and $\theta\,:\,\mathbb{R}\rightarrow G$ is a continuous homomorphism with a dense image.

$\textrm{C}(G)$ is the set of continuous functions on $G$ and $L^{p}(G),p\in[1,\infty)$ are Banach spaces. If $h\in\textrm{C}(G)$ then $f\mathrel{\mathop{:}}=h\circ\theta\in\textrm{U}(\mathbb{R})$ since by a theorem of Bochner [8] every sequence of translates of $f$ has a subsequence that converges uniformly. We call $h$ the lift of $f$ to $G.$ The Pontryagin dual [24] $\widehat{G}$ of a compact abelian group $G$ is the discrete group of continuous homomorphisms $\chi\,:\,G\rightarrow\mathbb{T}$ under pointwise multiplication. Bohr proved the existence of a compactification $(\mathbb{B},\theta)$ such that $\textrm{U}(\mathbb{R})=\{\,h\circ\theta\,:\,h\in\textrm{C}(\mathbb{B})\,\}.$ The group $\mathbb{B}$ is nonseparable and $\widehat{\mathbb{B}}$ is isomorphic to $\mathbb{R}_{\,d}\mathrel{\mathop{:}}=$ real numbers with the discrete topology.

Lemma 3

For every $f\in\textrm{U}(\mathbb{R})$ there exists a compactification $(G(f),\theta)$ and $h\in\textrm{C}(\mathbb{R})$ such that $f=h\circ\theta.$ The group $G(f)$ is separable.

Proof of Lemma 3 If $f\in\textrm{U}(\mathbb{R})$ is nonzero its spectrum $\Omega(f)$ is nonempty and countable so the product group $\mathbb{T}^{\,\Omega(f)}$ is compact and separable. The function $\theta\,:\,\mathbb{R}\rightarrow\mathbb{T}^{\,\Omega(f)}$ defined by $\theta(x)(\omega)\mathrel{\mathop{:}}=\chi_{\omega}(x)$ is a continuous homomorphism. Define $G(f)\mathrel{\mathop{:}}=\overline{\theta(\mathbb{R})}.$ Then $(G(f),\theta)$ is a compactification. The function $\widetilde{h}\,:\,\theta(\mathbb{R})\rightarrow\mathbb{C}$ defined by $\widetilde{h}(\theta(x)\mathrel{\mathop{:}}=f(x)$ is uniformly continuous so extends to a unique function $h\,:\,G\rightarrow\mathbb{C}$ and $f=h\circ\theta.$

Lemma 4

If $(G,\theta)$ is a compactification, $h\in\textrm{C}(G),$ $f=h\circ\theta,$ and $\log\circ|f|\in B^{p}(\mathbb{R}),$ then $\log\circ|h|\in L^{p}(G)$ and $\int_{G}|\log\circ|h|\,|^{p}=m(|\log\circ|f|\,|^{p}).$

Proof of Lemma 4 The theorem of averages ([3], p. 286) implies that

[TABLE]

The result follows from Lebesgue’s monotone convergence theorem since the sequence $|\log\circ|h|_{j}\,|^{p}$ is nondecreasing, converges pointwise to $|\log\circ|h|\,|^{p}$ pointwise and by (18) their integrals are uniformly bounded.

Definition 2

The Fourier transform $\mathfrak{F}\,:\,L^{1}(G)\rightarrow\ell^{\infty}(\widehat{G})$ is defined by $\mathfrak{F}(h)(\chi)\mathrel{\mathop{:}}=\int_{G}f\,\overline{\chi}.$

We define the spectrum $\Omega(h)\mathrel{\mathop{:}}=\hbox{support }\mathfrak{F}(h).$ The Hausdorff-Young theorem [15, 26] implies that the restrictions give bounded operators $\mathfrak{F}\,:\,L^{p}(G)\rightarrow\ell^{q}(\widehat{G})$ for $p\in[1,\infty)$ and $p^{-1}+q^{-1}=1.$

Definition 3

A compactification $(G,\theta)$ induces an injective homomorphism $\xi\,:\widehat{G}\rightarrow\mathbb{R},$ $\xi(\chi)\mathrel{\mathop{:}}=\omega$ where $\chi\circ\theta=\chi_{\omega},$ by which we will identity $\widehat{G}$ as a subset of $\mathbb{R}$ with the same archimedian order. Therefore if $h\in\textrm{C}(G)$ is the lift of $f\in\textrm{U}(\mathbb{R}),$ then $\Omega(h)=\Omega(f).$ The compactification gives Hardy spaces $H^{p}(G,\theta)\mathrel{\mathop{:}}=\{\,h\in L^{p}(G)\,:\,\Omega(h)\subset[0,\infty)\,\},p\in[1,\infty].$

Definition 4

A function $h\in H^{p}(G,\theta)$ is outer if $\int_{G}h\neq 0,$ $\log\circ|h|\in L^{1}(G),$ and

[TABLE]

A function $h\in H^{p}(G,\theta)$ is inner if $|h|=1.$

A polynomial $h$ is outer iff it has no zeros in the open unit disk since a formula of Jensen [16] gives $\int_{G}\log\circ\,|h|=\log|h(0)|-\sum_{h(\lambda)=0}\log^{-}(|\lambda|).$ Beurling [5] proved that a function $h\in H^{2}(\mathbb{T})$ admits a factorization $h=h_{o}\,h_{i},$ with $h_{o}$ outer and $h_{i}$ inner, iff $\log\circ\,|h|\in L^{1}(\mathbb{T}).$

Let $(G,\theta)$ be a compactification. If $h\in\textrm{C}(G)$ has a bounded spectrum $\Omega(h)\subset[0,\infty)$ then $f=h\circ\theta$ extends to an entire function $F$ bounded in the upper half plane. We observe that if $F$ has no zeros in the upper half plane, then $\chi_{-b(h)/2}F$ is the Ahiezer spectral factor [1] of the entire function $F(z)\overline{F(\overline{z})}.$

Conjecture 2

$h$* above is outer iff $F$ has no zeros in the open upper half plane.*

3 Mahler Measure and the Riemann Hypothesis

Definition 5

For $G$ is a compact abelian group the Mahler measure [22, 23] of $h\in L^{1}(G)$ is $M(h)\mathrel{\mathop{:}}=\exp\left(\,\int_{G}\log\circ|h|\,\right)\in[0,\infty).$ We also define $M^{\pm}(h)\mathrel{\mathop{:}}=\exp\left(\,\int_{G}\log^{\pm}\circ|h|\,\right).$

Since $M(h)=M^{+}(h)M^{-}(h)$ and $M^{+}(h)\in[\,1,\,\max\{1,||h||_{\infty}\}\,],$ it follows that $M(h)>0$ iff $\log^{-}\circ|h|\in L^{1}(G)$ and then $M^{-}(h)=\exp\left(-||\,\log^{-}\circ|h|\,||_{1}\,\right).$ Lemma 4 implies that this condition holds whenever $h\in\textrm{C}(G)$ is nonzero and $\Omega(h)$ is bounded.

Definition 6

For $N\in\mathbb{N},\,\Phi_{N}\mathrel{\mathop{:}}=$ product of the first $N$ cyclotomic polynomials.

Amoroso ([2], Theorem 1.3) proved that the Riemann Hypothesis is equivalent to

[TABLE]

Define $f_{N}\mathrel{\mathop{:}}=\Phi_{N}\circ\chi_{1}\in\textrm{U}(\mathbb{R})$ and define $J_{f_{N}}:(0,\infty)\rightarrow[0,1]$ by (1). Jensen’s formula implies that $M(\Phi_{N})=1$ therefore

[TABLE]

The bounds that we obtained for $J_{f}$ in (2) and (3) were exceptionally crude and totally inadequate to obtain (20). When deriving (3) for general polynomials we used the bound (8) $\Xi_{f,0,1,u}(v)=m(\{\,x\,:\,|f(x)|<u,|f^{(1)}(x)|<v\,\}\leq m(\{\,x\,:\,|f^{(1)}(x)|<v\,\}.$ Conjecture (1) was based on our intuition that a smaller upper bound holds. We suspect that much smaller upper bounds hold for specific sequences of polynomials as illustrated by the following examples. Construct sequences of height $1$ polynomials

[TABLE]

and $p_{n}\mathrel{\mathop{:}}=P_{n}\circ\chi_{1},q_{n}\mathrel{\mathop{:}}=Q_{n}\circ\chi_{1}.$ Both polynomials have maxima at $z=1,$ $||P_{n}||_{\infty}=n+1,$ Stirling’s approximation gives $||Q_{n}||_{\infty}\approx\sqrt{\pi\,n/2}$ for large $n,$ and for $u\in(0,1]$

[TABLE]

[TABLE]

Differences between these polynomials arise from their root discrepancy. Those of $P_{n}$ are nearly evenly spaced. Those of $Q_{n},$ all at $z=-1,$ have maximally discrepancy.

Conjecture 3

If $R_{n}$ is a polynomial with $n+1$ terms and height $H(R_{n})=1$ then $M^{-}(P_{n})\leq M^{-}(R_{n})\leq M^{-}(Q_{n}).$

The roots of $\Phi_{N}$ have the form $\exp(2\pi ia_{k}),k=1,...,\deg\Phi_{N}$ where $a_{k}$ are the Farey series consisting of rational numbers in $[0,1)$ whose denominators are $\leq N.$ Bounds on the discrepancy of the Farey series were shown by Franel [14] and by Landau [17] to imply the Riemann Hypothesis. The relationship between the discrepancy of roots of a polynomial and its coefficients, and the distributions of roots of entire functions have been extensively studied since the seminal paper by Erdös and Turán [12] and the extensive work by Levin and his school [20]. We suggest that investigation of the functions $\Xi_{f,\omega,k,v,u}$ in (4) and derived functions $K_{f}$ in (5) functions may further elucidate how the distribution of small values of polynomials and entire functions depend on their coefficients and roots.

Acknowledgment The author thanks Professor August Tsikh for insightful discussions.