aa r X i v : . [ m a t h . N T ] J u l On Mean Values of Dirichlet Polynomials
Michel WeberNovember 13, 2018
Abstract
We show the following general lower bound valid for any positive in-teger q , and arbitrary reals ϕ , . . . , ϕ N and non-negative reals a , . . . , a N , c q (cid:16) N X n =1 a n (cid:17) q ≤ T Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt. The object of this short Note is to prove the following lower bound
Theorem 1
For any positive integer q , there exists a constant c q , such that forany reals ϕ , . . . , ϕ N , any non-negative reals a , . . . , a N , and any T > , c q (cid:16) N X n =1 a n (cid:17) q ≤ T Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt. The result is no longer true for arbitrary reals a , . . . , a N as yields the case ϕ = . . . = ϕ N . It also follows that c (cid:16) N X n =1 a n (cid:17) / ≤ sup t ∈ R (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) . (1)In the case ϕ n = log n , it is known from [5] and [8] that for any ( a n )sup t ∈ R (cid:12)(cid:12)(cid:12) N − X n =0 a n n it (cid:12)(cid:12)(cid:12) ≥ α e β √ log N log log N √ N (cid:16) N − X n =0 | a n | (cid:17) (2)and for some ( a n )sup t ∈ R (cid:12)(cid:12)(cid:12) N − X n =0 a n n it (cid:12)(cid:12)(cid:12) ≤ α e β √ log N log log N √ N (cid:16) N − X n =0 | a n | (cid:17) , (3)with some universal constants α , α , β , β . Then (1) is better than (2) if forinstance a n = n − α , α > /
2, since e β √ log N log log N √ N (cid:16) N − X n =0 | a n | (cid:17) ∼ e β √ log N log log N ) N − α = o (1) ≪ (cid:16) N X n =1 a n (cid:17) / . L -case is related to well-known Ingham’s inequality [2]. We state thesharper form due to Mordell [7]: let 0 < ϕ < . . . < ϕ N and let γ be such thatmin
0. Then N sup n =1 | a n | ≤ KT Z T − T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) dt with T = πγ , (4)where K ≤ N sup n =1 | a n | ≤ lim sup T →∞ T Z T − T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) dt ≤ sup t ∈ R (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) , (5)a very familiar inequality in the theory of uniformly almost periodic functions.See also [1] where the more complicated inequality is established: | a n | ≤ Q n − j =0 cos( πϕ j ϕ n ) · Q Nn +1 cos( πϕ n ϕ j ) · sup | t |≤ π ( nϕn + P Nj = n +1 1 ϕj ) (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) . (6)In particular, if ϕ , . . . , ϕ N are linearly independent, and T is large enough,then b q (cid:16) N X n =1 a n (cid:17) q ≤ T Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt ≤ B q (cid:16) N X n =1 a n (cid:17) q , (7)holds for any nonnegative reals a , . . . , a N and b q , B q depend on q only.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
Let q be any positive integer. Let c , . . . , c N be complex numbers andnonnegative reals a , . . . , a N such that | c n | ≤ a n , n = 1 , . . . , N . Then for anyreals T, T with T > Z | t − T |≤ T (cid:12)(cid:12)(cid:12) N X n =1 c n e itϕ n (cid:12)(cid:12)(cid:12) q dt ≤ Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt. Proof.
Let K T ( t ) = K T ( | t | ) = (cid:0) − | t | /T ) χ {| t |≤ T } Observe that for any reals t, Ha ) K T ( t − H ) = (cid:0) − | t − H | /T ) χ {| t − H |≤ T } b ) χ {| t − H |≤ T } ≤ K T ( t − H ) + K T ( t − H + T ) + K T ( t − H − T ) c ) b K T ( u ) = 1 T (cid:0) sin T uu (cid:1) ≥ , for all real u. Suppose that | c n | ≤ a n for n = 1 , . . . , N . From (cid:16) N X n =1 c n e itϕ n (cid:17) q = X k + ... + k N = q (cid:16) q ! k ! . . . k N ! (cid:17) N Y n =1 c k n n e itk n ϕ n . (8)2nd (cid:12)(cid:12)(cid:12) N X n =1 c n e itϕ n (cid:12)(cid:12)(cid:12) q = X k ... + kN = qh ... + hN = q (cid:16) ( q !) k ! h ! . . . k N ! h N ! (cid:17) N Y n =1 c k n n c nh n e it ( k n − h n ) ϕ n we get Z R K T ( t − H ) (cid:12)(cid:12)(cid:12) N X n =1 c n e itϕ n (cid:12)(cid:12)(cid:12) q dt = X k ... + kN = qh ... + hN = q ( q !) k ! h ! . . . k N ! h N ! N Y n =1 c k n n c nh n Z R K T ( t − H ) e it P Nn =1 ( k n − h n ) ϕ n dt = X k ... + kN = qh ... + hN = q ( q !) k ! h ! . . . k N ! h N ! N Y n =1 c k n n c nh n Z R K T ( s ) e i ( s + H ) P Nn =1 ( k n − h n ) ϕ n ds = X k ... + kN = qh ... + hN = q ( q !) k ! h ! . . . k N ! h N ! N Y n =1 ( c n e iHϕ n ) k n ( c n e iHϕ n ) h n b K T (cid:16) N X n =1 ( k n − h n ) ϕ n (cid:17) ≤ X k ... + kN = qh ... + hN = q ( q !) k ! h ! . . . k N ! h N ! N Y n =1 a k n + h n n b K T (cid:16) N X n =1 ( k n − h n ) ϕ n (cid:17) = Z R K T ( t ) (cid:20) X k ... + kN = qh ... + hN = q ( q !) k ! h ! . . . k N ! h N ! N Y n =1 a k n + h n n e it P Nn =1 ( k n − h n ) ϕ n (cid:21) dt = Z R K T ( t ) (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt. Hence, if | c n | ≤ a n for n = 1 , . . . , N Z R K T ( t − H ) (cid:12)(cid:12)(cid:12) N X n =1 c n e itϕ n (cid:12)(cid:12)(cid:12) q dt ≤ Z R K T ( t ) (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt. (9)By applying Lemma 2 with H = 0 , T , − T , and using b), we get Z | t − T |≤ T (cid:12)(cid:12)(cid:12) N X n =1 c n e itϕ n (cid:12)(cid:12)(cid:12) q dt ≤ Z R (cid:16) K T ( t − T ) + K T ( t − T + T ) + K T ( t − T − T ) (cid:17)(cid:12)(cid:12)(cid:12) N X n =1 c n e itϕ n (cid:12)(cid:12)(cid:12) q dt ≤ Z R K T ( t ) (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt ≤ Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt. (10)3he proof of Theorem 1 is now achieved as follows. First recall the Khintchin-Kahane inequalities [4]. Let { ε i , ≤ i ≤ N } be independent Rademacherrandom variables, thus satisfying P { ε i = ± } = 1 /
2, if (Ω , A , P ) denotes theunderlying basic probability. Then for any 0 < p < ∞ , there exist positive finiteconstants c p , C p depending on p only, such that for any sequence { a i , ≤ i ≤ N } of real numbers c p (cid:16) N X i =1 a i (cid:17) / ≤ (cid:13)(cid:13)(cid:13) N X i =1 a i ε i (cid:13)(cid:13)(cid:13) p ≤ C p (cid:16) N X i =1 a i (cid:17) / . (11)This remains true for complex a n . If a n = α n + iβ n , then (cid:13)(cid:13)(cid:13) N X j =1 a j ε j (cid:13)(cid:13)(cid:13) pp = E (cid:12)(cid:12)(cid:12) N X j =1 α j ε j + i N X j =1 β j ε j (cid:12)(cid:12)(cid:12) p = E (cid:16)(cid:12)(cid:12)(cid:12) N X j =1 α j ε j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) N X j =1 β j ε j (cid:12)(cid:12)(cid:12) (cid:17) p/ ≤ ( p/ − (cid:16) E (cid:12)(cid:12)(cid:12) N X j =1 α j ε j (cid:12)(cid:12)(cid:12) p + E (cid:12)(cid:12)(cid:12) N X j =1 β j ε j (cid:12)(cid:12)(cid:12) p (cid:17) , where we have denoted by E the corresponding expectation symbol. Thus, since √ A + √ B ≤ p A + B ), A, B ≥ (cid:13)(cid:13)(cid:13) N X j =1 a j ε j (cid:13)(cid:13)(cid:13) p ≤ / − /p C p h(cid:0) N X j =1 α j (cid:1) / + (cid:0) N X j =1 β j (cid:1) / i ≤ − /p C p (cid:16) N X j =1 ( α j + β j ) (cid:17) / = C ′ p (cid:16) N X j =1 | a j | (cid:17) / . Conversely, from (cid:13)(cid:13)(cid:13) N X j =1 a j ε j (cid:13)(cid:13)(cid:13) pp = E (cid:16)(cid:12)(cid:12)(cid:12) N X j =1 α j ε j (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) N X j =1 β j ε j (cid:12)(cid:12)(cid:12) (cid:17) p/ ≥ max (cid:16) E (cid:12)(cid:12)(cid:12) N X j =1 α j ε j (cid:12)(cid:12)(cid:12) p , E (cid:12)(cid:12)(cid:12) N X j =1 β j ε j (cid:12)(cid:12)(cid:12) p (cid:17) , we get (cid:13)(cid:13)(cid:13) N X j =1 a j ε j (cid:13)(cid:13)(cid:13) p ≥ max (cid:16)(cid:13)(cid:13) N X j =1 α j ε j (cid:13)(cid:13) p , (cid:13)(cid:13) N X j =1 β j ε j (cid:13)(cid:13) p (cid:17) ≥ c p max (cid:16)(cid:0) N X j =1 | α j | (cid:1) / , (cid:0) N X j =1 | β j | (cid:1) / (cid:17) ≥ c p (cid:16) N X j =1 ( | α j | + | β j | ) (cid:17) / = c ′ p (cid:16) N X j =1 | a j | (cid:17) / . Now choose c n = ε n a n . Taking expectation in inequality of Lemma 2.1, andusing Fubini’s Theorem, gives Z | t |≤ T E (cid:12)(cid:12)(cid:12) N X n =1 ε n a n e itϕ n (cid:12)(cid:12)(cid:12) q ≤ Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt. (12)4y (11) we have c q (cid:16) N X n =1 a n (cid:17) q ≤ E (cid:12)(cid:12)(cid:12) N X n =1 ε n a n e itϕ n (cid:12)(cid:12)(cid:12) q ≤ C q (cid:16) N X n =1 a n (cid:17) q . (13)By reporting 2 T c q (cid:16) N X n =1 a n (cid:17) q ≤ Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 a n e itϕ n (cid:12)(cid:12)(cid:12) q dt, (14)which proves our claim. We shall deduce from Theorem 1 the following lower bound.
Corollary 3
For every N , T and νc ν log ν N ≤ T Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 n + it (cid:12)(cid:12)(cid:12) ν dt. In relation with this is Ramachandra’s well-known lower bound (see [3] section9.5, to which we also refer for the estimates used in the proof) c ν (log T ) ν ≤ T Z | t |≤ T (cid:12)(cid:12) ζ ( 12 + it ) (cid:12)(cid:12) ν dt. (15) Proof.
Apply Theorem 1 with q = 2 to the sum (cid:16) N X n =1 n + it (cid:17) ν := N ν X m =1 b m m + it , where b m = (cid:8) ( n j ) j ≤ ν ; n j ≤ N : m = Y j ≤ ν n j (cid:9) . Thus for all N and Tc ν N ν X m =1 b m m ≤ T Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 n + it (cid:12)(cid:12)(cid:12) ν dt. But if m ≤ N , b m = d ν ( m ) where d ν ( m ) denotes the number of representationsof m as a product of ν factors, and we know that X m ≤ x d ν ( m ) m = ( C ν + o (1)) log ν x. Thus N ν X m =1 b m m ≥ N X m =1 b m m ≥ c ν log ν N Henceforth c ν log ν N ≤ T Z | t |≤ T (cid:12)(cid:12)(cid:12) N X n =1 n + it (cid:12)(cid:12)(cid:12) ν dt. cknowlegments. I thank Professor Aleksandar Ivi´c for useful remarks.
References [1] Binmore K.G. [1966]:
A trigonometric inequality , J. London Math. Soc. , 693–696.[2] Ingham A.E. [1950]: A further note on trigonometrical inequalities , Proc.Cambridge Philos. Soc. , 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 Monographs , American Math. Soc.[5] Konyagin S.V., Queff´elec H. [2001/2002] The translation in the theory ofDirichlet series , Real Anal. Exchange (1), 155–176.[6] Montgomery H. [1993]:
Ten lectures on the interface between analyticnumber theory and harmonic analysis , Conference Board of the Math. Sci-ences, Regional Conference Series in Math. .[7] Mordell I.J. [1957]:
On Ingham’s trigonometric inequality , Illinois J.Math. , 214–216.[8] Queff´elec H. [1995] H. Bohr’s vision of ordinary Dirichlet series; old andnew results , J. Analysis , p.43-60.[9] Ramachandra K. [1995] On the Mean-Value and Omega-Theorems for theRiemann 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 rueRen´e Descartes, 67084 Strasbourg Cedex, France.E-mail: [email protected]@math.u-strasbg.fr