On mean values of Dirichlet polynomials

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On mean values of Dirichlet polynomials

Michel Weber

To cite this version:

Michel Weber. On mean values of Dirichlet polynomials. Mathematical Inequalities & Applications,

2011, 14 (3), pp.529-534. �10.7153/mia-14-45�. �hal-01279465�

arXiv:0907.4767v1 [math.NT] 28 Jul 2009

On Mean Values of Dirichlet Polynomials

Michel Weber July 29, 2009

Abstract

We show the following general lower bound valid for any positive in- tegerq, and arbitrary realsϕ₁, . . . , ϕN and non-negative realsa₁, . . . , aN,

cq

X^N

n=1

a²n

^q

≤ 1 2T

Z

|t|≤T

XN

n=1

ane^itϕn

2q

dt.

1 Main Result

The object of this short Note is to prove the following lower bound

Theorem 1 For any positive integerq, there exists a constantcq, such that for any realsϕ1, . . . , ϕN, any non-negative reals a1, . . . , aN, and anyT >0,

cq

X^N

n=1

a²_nq

≤ 1 2T

Z

|t|≤T

XN n=1

ane^itϕn

2q

dt.

The result is no longer true for arbitrary reals a1, . . . , aN as yields the case ϕ1=. . .=ϕN. It also follows that

cX^N

n=1

a²_n1/2

≤sup

t∈R

XN n=1

ane^itϕn

. (1) In the caseϕn= logn, it is known from [5] and [8] that for any (an)

sup

t∈R

NX−1 n=0

ann^it

≥α1e^β1√

logNlog logN

√N

^NX⁻¹

n=0

|an|

(2) and for some (an)

sup

t∈R

N−1X

n=0

ann^it

≤α2e^β2√

logNlog logN

√N

^NX⁻¹

n=0

|an|

, (3)

with some universal constantsα1, α2, β1, β2. Then (1) is better than (2) if for instancean =n^−α,α >1/2, since

e^β1√

logNlog logN

√N

^N−1X

n=0

|an|

∼e^β1√

logNlog logN)N^12−α=o(1)≪X^N

n=1

a²_n1/2

.

TheL¹-case is related to well-known Ingham’s inequality [2]. We state the sharper form due to Mordell [7]: let 0< ϕ1< . . . < ϕN and letγ be such that

1<n≤Nmin ϕn−ϕn−1≥γ >0. Then

supN

n=1|an| ≤ K T

Z T

−T

XN n=1

ane^itϕn

dt withT =π

γ, (4)

whereK≤1.

Further with no restriction, one always have supN

n=1|an| ≤lim sup

T→∞

1 2T

Z T

−T

XN n=1

ane^itϕn

dt≤sup

t∈R

XN n=1

ane^itϕn

, (5)

a very familiar inequality in the theory of uniformly almost periodic functions.

See also [1] where the more complicated inequality is established:

|an| ≤ 1 Qn−1

j=0 cos(^πϕ_2ϕ^j

n)·QN

n+1cos(^πϕ_2ϕⁿ

j)· sup

|t|≤^π₂(_ϕnⁿ +PN j=n+1

1 ϕj)

XN n=1

ane^itϕn .

(6) In particular, if ϕ1, . . . , ϕN are linearly independent, and T is large enough, then

bq

X^N

n=1

a²_nq

≤ 1 2T

Z

|t|≤T

XN n=1

ane^itϕn

2q

dt≤Bq

X^N

n=1

a²_nq

, (7)

holds for any nonnegative realsa1, . . . , aN andbq,Bq depend onqonly.

The proof of Theorem 1 relies upon the following lemma, which just gen- eralizes a useful majorization argument ([6], p.131) to arbitrary even powers.

Lemma 2 Letqbe any positive integer. Letc1, . . . , cN be complex numbers and nonnegative reals a1, . . . , aN such that |cn| ≤an, n= 1, . . . , N. Then for any reals T, T0 with T >0

Z

|t−T0|≤T

XN n=1

cne^itϕn

2q

dt≤3 Z

|t|≤T

XN n=1

ane^itϕn

2q

dt.

Proof. Let

KT(t) =KT(|t|) = 1− |t|/T)χ{|t|≤T}

Observe that for any realst, H

a) KT(t−H) = 1− |t−H|/T)χ{|t−H|≤T}

b) χ{|t−H|≤T}≤KT(t−H) +KT(t−H+T) +KT(t−H−T) c) KbT(u) = 1

T

sinT u u

2

≥0, for all realu.

Suppose that|cn| ≤an forn= 1, . . . , N. From X^N

n=1

cne^itϕnq

= X

k1+...+kN=q

q!

k1!. . . kN! Y^N

n=1

c^k_nⁿe^itknϕn. (8)

and

XN n=1

cne^itϕn

2q

= X

k1 +...+kN=q h1 +...+hN=q

(q!)² k1!h1!. . . kN!hN!

Y^N

n=1

c^k_nⁿcnhne^{it(kn−hn)ϕn}

we get Z

R

KT(t−H)

XN n=1

cne^itϕn

2q

dt

= X

k1 +...+kN=q h1 +...+hN=q

(q!)² k1!h1!. . . kN!hN!

YN n=1

c^k_nⁿcnhn

Z

R

KT(t−H)e^itPN

n=1(kn−hn)ϕndt

= X

k1 +...+kN=q h1 +...+hN=q

(q!)² k1!h1!. . . kN!hN!

YN n=1

c^k_nⁿcnhn

Z

R

KT(s)e^i(s+H)PN

n=1(kn−hn)ϕnds

= X

k1 +...+kN=q h1 +...+hN=q

(q!)² k1!h1!. . . kN!hN!

YN n=1

(cne^iHϕn)^kn(cne^iHϕn)^hnKbT

X^N

n=1

(kn−hn)ϕn

≤ X

k1 +...+kN=q h1 +...+hN=q

(q!)² k1!h1!. . . kN!hN!

YN n=1

a^kn^n+hnKbT

X^N

n=1

(kn−hn)ϕn

= Z

R

KT(t)

X

k1 +...+kN=q h1 +...+hN=q

(q!)² k1!h1!. . . kN!hN!

YN n=1

a^k_n^n+hne^it PN

n=1(kn−hn)ϕn

dt

= Z

R

KT(t)

XN n=1

ane^itϕn

2q

dt.

Hence, if|cn| ≤an forn= 1, . . . , N Z

R

KT(t−H)

XN n=1

cne^itϕn

2q

dt≤ Z

R

KT(t)

XN n=1

ane^itϕn

2q

dt. (9)

By applying Lemma 2 withH = 0, T0,−T0, and using b), we get Z

|t−T0|≤T

XN n=1

cne^itϕn

2q

dt

≤ Z

R

KT(t−T0) +KT(t−T0+T) +KT(t−T0−T)

XN n=1

cne^itϕn

2q

dt

≤ 3 Z

R

KT(t)

XN n=1

ane^itϕn

2q

dt≤3 Z

|t|≤T

XN n=1

ane^itϕn

2q

dt. (10)

The proof of Theorem 1 is now achieved as follows. First recall the Khintchin- Kahane inequalities [4]. Let {εi,1 ≤ i ≤ N} be independent Rademacher random variables, thus satisfying P{εi =±1} = 1/2, if (Ω,A,P) denotes the underlying basic probability. Then for any 0< p <∞, there exist positive finite constantscp,Cpdepending onponly, such that for any sequence{ai,1≤i≤N} of real numbers

cp

X^N

i=1

a²_i1/2

≤

XN i=1

aiεi

p≤Cp

X^N

i=1

a²_i1/2

. (11)

This remains true for complexan. Ifan=αn+iβn, then

XN j=1

ajεj

p

p = E

XN j=1

αjεj+i XN j=1

βjεj

p

=E

XN j=1

αjεj

2

+

XN j=1

βjεj

2p/2

≤ 2^(p/2)−1 E

XN j=1

αjεj

p

+E

XN j=1

βjεj

p ,

where we have denoted byEthe corresponding expectation symbol. Thus, since

√A+√ B≤p

2(A+B),A, B≥0,

XN j=1

ajεj

p ≤ 2^1/2−1/pCp

h X^N

j=1

α²_j1/2

+ XN j=1

β_j²1/2i

≤ 2^1−1/pCp

X^N

j=1

(α²_j+β²_j)1/2

=C_p^′X^N

j=1

|aj|²1/2

. Conversely, from

XN j=1

ajεj

p

p = E

XN j=1

αjεj

2

+

XN j=1

βjεj

2p/2

≥ max E

XN j=1

αjεj

p

,E

XN j=1

βjεj

p , we get

XN j=1

ajεj

p ≥ max

XN j=1

αjεj

p,

XN j=1

βjεj

p

≥ cpmax X^N

j=1

|αj|²1/2

, XN j=1

|βj|²1/2

≥ cp

2 X^N

j=1

(|αj|²+|βj|²)1/2

=c^′_pX^N

j=1

|aj|²1/2

. Now choose cn =εnan. Taking expectation in inequality of Lemma 2.1, and using Fubini’s Theorem, gives

Z

|t|≤T

E

XN n=1

εnane^itϕn

2q

≤3 Z

|t|≤T

XN n=1

ane^itϕn

2q

dt. (12)

By (11) we have cq

X^N

n=1

a²_nq

≤E

XN n=1

εnane^itϕn

2q

≤Cq

X^N

n=1

a²_nq

. (13)

By reporting

2T cq

X^N

n=1

a²_nq

≤3 Z

|t|≤T

XN n=1

ane^itϕn

2q

dt, (14)

which proves our claim.

2 Application

We shall deduce from Theorem 1 the following lower bound.

Corollary 3 For everyN,T andν cνlog^ν2N ≤ 1

2T Z

|t|≤T

XN n=1

1 n^12+it

2ν

dt.

In relation with this is Ramachandra’s well-known lower bound (see [3] section 9.5, to which we also refer for the estimates used in the proof)

cν(logT)^ν2≤ 1 2T

Z

|t|≤T

ζ(1 2+it)

^2νdt. (15) Proof. Apply Theorem 1 with q= 2 to the sum

X^N

n=1

1 n^12+it

ν

:=

N^ν

X

m=1

bm

m^12+it, where

bm= #

(nj)j≤ν; nj≤N :m=Y

j≤ν

nj . Thus for allN andT

cν N^ν

X

m=1

b²_m m ≤ 1

2T Z

|t|≤T

XN n=1

1 n^12+it

2ν

dt.

But ifm≤N,bm=dν(m) wheredν(m) denotes the number of representations ofmas a product ofν factors, and we know that

X

m≤x

d²_ν(m)

m = (Cν+o(1)) log^ν2x.

Thus XN^ν

m=1

b²_m

m ≥

XN m=1

b²_m

m ≥cνlog^ν2N Henceforth

cνlog^ν2N ≤ 1 2T

Z

|t|≤T

XN n=1

1 n^12+it

2ν

dt.

Acknowlegments. I thank Professor Aleksandar Ivi´c for useful remarks.

References

[1] Binmore K.G. [1966]: A trigonometric inequality, J. London Math. Soc.

41, 693–696.

[2] Ingham A.E. [1950]: A further note on trigonometrical inequalities, Proc.

Cambridge Philos. Soc.46, 535-537.

[3] Ivi´c A. [1985]The Riemann Zeta-function, Wiley-Interscience Publication, J. Wiley&Sons, New-York.

[4] Kashin B.S., Saakyan A.A. [1989]Orthogonal Series, Translations of Math- ematical Monographs75, American Math. Soc.

[5] Konyagin S.V., Queff´elec H. [2001/2002]The translation ¹₂ in the theory of Dirichlet series, Real Anal. Exchange27(1), 155–176.

[6] Montgomery H.[1993]: Ten lectures on the interface between analytic number theory and harmonic analysis, Conference Board of the Math. Sci- ences, Regional Conference Series in Math.84.

[7] Mordell I.J. [1957]: On Ingham’s trigonometric inequality, Illinois J.

Math.1, 214–216.

[8] Queff´elec H. [1995] H. Bohr’s vision of ordinary Dirichlet series; old and new results, J. Analysis3, p.43-60.

[9] Ramachandra K. [1995]On the Mean-Value and Omega-Theorems for the Riemann Zeta-Function, Tata Institute of Fundamental Research, Bombay, Springer Verlag Berlin, Heidelberg, New-York, Tokyo, vii+167p.

Michel Weber, Math´ematique (IRMA), Universit´e Louis-Pasteur et C.N.R.S., 7 rue Ren´e Descartes, 67084 Strasbourg Cedex, France.

E-mail: weber@math.u-strasbg.fr