aa r X i v : . [ m a t h . N T ] D ec CAUCHY MEANS OF DIRICHLET POLYNOMIALS
MICHEL J. G. WEBER
Abstract.
We study Cauchy means of Dirichlet polynomials Z R (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) . These integrals were investigated when q = 1 , σ = 1 , s = 1 / q ≥ σ ≥ s >
0. Wecomplete Wilf’s approach by relating it with other approaches (having notably connection withBrownian motion), allowing simple proofs, and also prove new results.
1. Introduction and Main Results.
In a quite inspiring paper [8], Wilf has considered integral operators associated with homoge-neous, nonnegative kernels K ( x, y ) and applied his results to Dirichlet series. Consider for instancethe kernel K ( x, y ) = max( x, y ) − . It has Mellin transform F ( s ) = Z ∞ t − s K ( t, t = 1 s + 11 − s ,s = σ + it , which is invertible on the critical line. As further K ( x, y ) is symmetric and decreasing,it is well-known in this case that the spectral theory of K ( x, y ) depends on the behavior of theMellin transform of K ( t,
1) along the critical line.If x , . . . , x N are complex numbers, then ([8], Theorem 3)(1.1) N X n,m =1 ¯ x n K ( n, m ) x m = 12 π Z R F ( 12 + it ) (cid:12)(cid:12)(cid:12) N X n =1 x n n + it (cid:12)(cid:12)(cid:12) d t ≤ F ( 12 ) N X j =1 | x n | . The last inequality follows from Widom’s eigenvalue estimate ([8], Theorem 2). Wilf has shownthat (1.1) holds for the class H of kernels K such that K ( x, y ) ≥ x, y nonnegative, and isfurther symmetric, decreasing and homogeneous of degree −
1: for every α > K ( αx, αy ) = α − K ( x, y ) ∀ x > , ∀ y > . In the case considered, (1.1) implies that(1.3) Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n + it (cid:12)(cid:12)(cid:12) d t + t ≤ π N X n =1 | x n | . Taking x n = n − / yields in particular the following nice bound ([8], (17))(1.4) Z R (cid:12)(cid:12)(cid:12) N X n =1 n it (cid:12)(cid:12)(cid:12) d t + t ≤ π N X n =1 n ≤ C log N. That inequality is in turn two-sided and this can be showed without appealing to Mellin trans-form nor Widom’s eigenvalue estimate. The purpose of this Note is to first relate Wilf’s approachwith other approaches allowing simple proofs, and next, to develop more some parts and provenew results. The above integrals are Cauchy means on the real line of Dirichlet polynomials, and
Key words and phrases.
Dirichlet polynomials, Cauchy density, arctangent density, mean-value, Mellin trans-form, spectral theory, homogeneous kernels, Brownian motion. admit an exact formulation. This is in contrast with usual mean-value of Dirichlet polynomials,with respect to measures χ [0 ,T ] ( t )d t/T , where an error term always occurs due to the fact that(1.5) Z T (cid:16) mn (cid:17) it d t = ( T if m = n O m,n (1) otherwise . Both means are in turn strongly related. Cauchy means of Dirichlet polynomials are part ofthe theory of Dirichlet polynomials for various weights and it is expected that their study will givenew insight into properties of general Dirichlet polynomials. We refer for instance to the recentworks of Lubinsky [3, 4], [which we discovered while this work was much advanced].As the weight functions in turn represent a sampling of the parameter t , the properties of theweighted Dirichlet approximating polynomials can be used to study the behavior of the Riemannzeta function ζ ( σ + it ) along the critical line σ = 1 /
2. A (rather) elaborated application of this,in the case of the Cauchy density, can be found in Lifshits and Weber [2].We begin with giving proofs of (1.3), (1.4) without appealing to spectral theory (Widom’seigenvalue estimate).
We start with an elementary lemma.
Lemma 1.1.
Let s ∈ R + and x , . . . , x N , y , . . . , y N be complex numbers. We have Z R (cid:12)(cid:12)(cid:12) M X ν =1 y ν ν ist (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = M X µ,ν =1 N X m,n =1 y µ x m y ν x n (cid:16) nν ∧ mµnν ∨ mµ (cid:17) s Moreover, lim s →∞ Z R (cid:12)(cid:12)(cid:12) M X ν =1 y ν ν ist (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = X ≤ µ,ν ≤ M ≤ m,n ≤ Nnν = mµ y µ x m y ν x n . Remark . The last assertion implies thatlim s →∞ Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = N X n =1 | x n | lim s →∞ Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = X ≤ µ,ν ≤ N ≤ m,n ≤ Nnν = mµ x µ x m x ν x n . Taking x n = n − σ yields,lim s →∞ Z R (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = N X n =1 n σ lim s →∞ Z R (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = X ≤ µ,ν ≤ N ≤ m,n ≤ Nnν = mµ µmνn ) σ . And in particular, by using Ayyad, Cochrane and Zheng estimate [1], Theorem 3,lim s →∞ Z R (cid:12)(cid:12)(cid:12) N X n =1 n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = n ≤ µ, ν ≤ N, ≤ m, n ≤ N : nν = mµ o = 12 π N log N + CN + O (cid:0) N / log / N (cid:1) , where C = π (12 γ − ( π ζ ′ (2) − − γ is Euler’s constant and ζ ′ (2) = P ∞ n =1 log nn . Proof.
From the relation e −| ϑ | = R R e iϑt d tπ ( t +1) , it follows that (cid:16) nm (cid:17) s = Z R n ist m − ist d tπ ( t + 1) ( m ≥ n ) . (1.6) AUCHY MEANS OF DIRICHLET POLYNOMIALS 3
Thus Z R (cid:12)(cid:12)(cid:12) M X ν =1 y ν ν ist (cid:12)(cid:12)(cid:12) n ist m − ist d tπ ( t + 1) = M X µ,ν =1 y µ y ν Z R nν ) ist ( mµ ) − ist d tπ ( t + 1)= M X µ,ν =1 y µ y ν (cid:16) nν ∧ mµnν ∨ mµ (cid:17) s . Consequently Z R (cid:12)(cid:12)(cid:12) M X ν =1 y ν ν ist (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = M X µ,ν =1 N X m,n =1 y µ y ν Z R nν ) ist ( mµ ) − ist d tπ ( t + 1)= M X µ,ν =1 N X m,n =1 y µ x m y ν x n (cid:16) nν ∧ mµnν ∨ mµ (cid:17) s . The second assertion follows easily. Let δ = max ≤ µ,ν ≤ M ≤ m,n ≤ Nnν = mµ (cid:16) nν ∧ mµnν ∨ mµ (cid:17) s . Then 0 < δ <
1. And the conclusion follows from Z R (cid:12)(cid:12)(cid:12) M X ν =1 y ν ν ist (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = X ≤ µ,ν ≤ M ≤ m,n ≤ Nnν = mµ y µ x m y ν x n + ( M N ) O ( δ s ) . (cid:3) To recover (1.4) and also to prove the corresponding lower bound, take x n = n − , M = 1 = y and s = 1 /
2. We get12 π Z R (cid:12)(cid:12)(cid:12) N X n =1 n iθ (cid:12)(cid:12)(cid:12) d θ + θ = N X m,n =1 m ∧ n ) / ( m ∨ n ) / = N X n =1 n + 2 N X n =1 n / N X m = n +1 m / ≤ C (cid:16) N X n =1 n (cid:17) ≤ C log N, which is (1.4). And obviously,12 π Z R (cid:12)(cid:12)(cid:12) N X n =1 n iθ (cid:12)(cid:12)(cid:12) d θ + θ ≥ N/ X n =1 n / N X m = n +1 m / ≥ C N X n =1 n ≥ C log N. Let W = { W ( t ) , t ≥ } be standard one-dimensional Brownian motion issued from 0 at time t = 0 and with underlying probability space(Ω , A , P ). Then(1.7) K ( s, t ) = ( s ∧ t ) st = E (cid:16) W ( s ) s W ( t ) t (cid:17) and E (cid:16) W ( s ) √ s W ( t ) √ t (cid:17) = (cid:16) s ∧ ts ∨ t (cid:17) / . This allows to interpret these integrals as Brownian sums, and by using the independence of theincrements of W , to find another convenient reformulation. Lemma 1.3.
For any real s ≥ , Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = E (cid:12)(cid:12)(cid:12) N X n =1 x n W ( n s ) n s (cid:12)(cid:12)(cid:12) = N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) N X µ = j x µ µ s (cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = E (cid:12)(cid:12)(cid:12) N X n,ν =1 x ν x n W (( nν ) s )( nν ) s (cid:12)(cid:12)(cid:12) = N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) X ≤ n,ν ≤ Nnν ≥ j x ν x n ( nν ) s (cid:12)(cid:12)(cid:12) . Proof.
The first equality follows from Lemma 1.1 and (1.7). As to the second one, write W (( nν ) s ) = P nνj =0 g j , where g j = W ( j s ) − W (( j − s ), j ≥
1, we also have Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = N X n,ν =1 N X m,µ =1 x µ x m x ν x n (cid:16) nν ∧ mµnν ∨ mµ (cid:17) s = N X n,ν =1 N X m,µ =1 x µ x m x ν x n E (cid:16) W (( nν ) s )( nν ) s W (( mµ ) s )( mµ ) s (cid:17) = E (cid:12)(cid:12)(cid:12) N X n,ν =1 x ν x n W (( nν ) s )( nν ) s (cid:12)(cid:12)(cid:12) = E (cid:12)(cid:12)(cid:12) N X n,ν =1 x ν x n ( nν ) s nν X j =1 g j (cid:12)(cid:12)(cid:12) = E (cid:12)(cid:12)(cid:12) N X j =1 g j X ≤ n,ν ≤ Nnν ≥ j x ν x n ( nν ) s (cid:12)(cid:12)(cid:12) = N X j =1 (cid:12)(cid:12)(cid:12) X ≤ n,ν ≤ Nnν ≥ j x ν x n ( nν ) s (cid:12)(cid:12)(cid:12) E g j = N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) X ≤ n,ν ≤ Nnν ≥ j x ν x n ( nν ) s (cid:12)(cid:12)(cid:12) . (cid:3) We now need a technical lemma.
Lemma 1.4.
For any s > and complex numbers x j , j = 1 , . . . , N , N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) N X µ = j x µ µ s (cid:12)(cid:12)(cid:12) ≤ C s P Nµ =1 | x µ | µ / − s if 0 < s < / ,C P Nµ =1 | x µ | µ log µ if s = 1 / C s P Nµ =1 | x µ | µ if s > / . Proof.
Let y µ = x µ /µ s − . By H¨older’s inequality, N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) N X µ = j x µ µ s (cid:12)(cid:12)(cid:12) = N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) N X µ = j y µ µ (cid:12)(cid:12)(cid:12) (writing µ = µ / .µ / ) ≤ N X j =1 ( j s − ( j − s ) (cid:16) N X µ = j µ / (cid:17)(cid:16) N X µ = j | y µ | µ / (cid:17) ≤ C s N X j =1 ( j s − ( j − s ) j / (cid:16) N X µ = j | y µ | µ / (cid:17) ≤ C s N X µ =1 | y µ | µ / X j ≤ µ j s − / . If 0 < s < /
4, it follows that N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) N X µ = j x µ µ s (cid:12)(cid:12)(cid:12) ≤ C s N X µ =1 | y µ | µ / = C s N X µ =1 | x µ | µ / − s . AUCHY MEANS OF DIRICHLET POLYNOMIALS 5 If s > / N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) N X µ = j x µ µ s (cid:12)(cid:12)(cid:12) ≤ C s N X µ =1 | y µ | µ / X j ≤ µ j s − / ≤ C s N X µ =1 | y µ | µ / µ s − / = C s N X µ =1 | x µ | µ / − s µ s − / = C s N X µ =1 | x µ | µ. And if s = 1 / N X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) N X µ = j x µ µ s (cid:12)(cid:12)(cid:12) ≤ C N X µ =1 | y µ | µ / X j ≤ µ j − ≤ C N X µ =1 | y µ | log µµ / = C N X µ =1 | x µ | µ log µ. (cid:3) Indicate now how to deduce (1.3). By taking s = 1 / x j = z j /j / we get in particular N X k =1 (cid:12)(cid:12)(cid:12) N X j = k z j j (cid:12)(cid:12)(cid:12) ≤ C N X j =1 | z j | , hence by Lemma 1.3, Z R (cid:12)(cid:12)(cid:12) N X n =1 z n n (1+ it ) (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = N X k =1 (cid:12)(cid:12)(cid:12) N X j = k z j j (cid:12)(cid:12)(cid:12) ≤ C N X j =1 | z j | . Making the variable change t = 2 θ , gives Z R (cid:12)(cid:12)(cid:12) N X n =1 z n n + iθ (cid:12)(cid:12)(cid:12) d θπ ( θ + ) ≤ C N X j =1 | z j | , which is (1.3) up to the value of the constant. One can deduce similar estimates for integrals of power four. C (log N ) ≤ Z R (cid:12)(cid:12)(cid:12) N X n =1 n it/ (cid:12)(cid:12)(cid:12) d tπ ( t + 1) ≤ C (log N ) . Take s = 1 / x n = 1 /n . Then Z R (cid:12)(cid:12)(cid:12) N X n =1 n it/ (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = N X j =1 (cid:12)(cid:12)(cid:12) X ≤ n,ν ≤ Nnν ≥ j nν ) (cid:12)(cid:12)(cid:12) . Next X ≤ n,ν ≤ Nnν ≥ j nν ) = X j Further, for j ≤ N/ X ≤ n,ν ≤ Nnν ≥ j nν ) ≥ X ≤ n ≤ j n X ≤ ν ≤ Nν ≥ j/n ν ≥ C X ≤ n ≤ j n (cid:16) nj (cid:17) = C log jj , and Z R (cid:12)(cid:12)(cid:12) N X n =1 n it (cid:12)(cid:12)(cid:12) d tπ ( t + 1) ≥ C (log N ) . L . It is natural to consider the (Hilbert) space L consisting with allBorel-measurable functions f : R → C such that k f k = Z R | f ( t ) | d tπ ( t + 1) < ∞ . That question was recently investigated by Lubinsky in [3]. Let λ = 0 and 1 = λ < λ < . . . withlim k →∞ λ k = ∞ . Applying the Gram-Schmidt process to { λ − itn , n ≥ } , produces the sequence oforthonormal Dirichlet polynomials φ n ( t ) = λ − itn − λ − itn q λ n − λ n − , n = 1 , , . . . Let F ( t ) = P ∞ n =1 a n λ − itn where { a n , n ≥ } ⊂ C and let s > 0. Recall Th. 1.1 in [3]. Assumethat the series ∞ X n =1 ( λ sk − λ sn − ) (cid:12)(cid:12)(cid:12) ∞ X n = k a n λ sn (cid:12)(cid:12)(cid:12) converges. Then F ( s. ) ∈ L and(1.8) Z R | F ( st ) | d tπ (1 + t ) = ∞ X n =1 ( λ sk − λ sn − ) (cid:12)(cid:12)(cid:12) ∞ X n = k a n λ sn (cid:12)(cid:12)(cid:12) . Further, F ( s. ) is the limit in L of some (explicited) subsequence of its partial sums.Consequently, in Lemma 1.3, we also have that Z R (cid:12)(cid:12)(cid:12) ∞ X n =1 x n n ist (cid:12)(cid:12)(cid:12) d tπ ( t + 1) = ∞ X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) ∞ X µ = j x µ µ s (cid:12)(cid:12)(cid:12) = E (cid:12)(cid:12)(cid:12) ∞ X n =1 x n W ( n s ) n s (cid:12)(cid:12)(cid:12) (1.9)provided that the Brownian series P ∞ n =1 x n W ( n s ) n s converges in L ( P ).New sufficient conditions for F to belong to L can further easily be derived from Lemma 1.4.More precisely, Corollary 1.5. Let F ( t ) = P ∞ n =1 x n n − it where x n ≥ and let s > . A sufficient condition for F ( s. ) ∈ L is P ∞ µ =1 x µ µ / − s < ∞ if 0 < s < / , P ∞ µ =1 x µ µ log µ < ∞ if s = 1 / , P ∞ µ =1 x µ µ < ∞ if s > / . Proof. Under either of these conditions, the corresponding series ∞ X j =1 ( j s − ( j − s ) (cid:16) ∞ X µ = j x µ µ s (cid:17) is convergent, since for instance if s > / 4, by Lemma 1.4, for all N ≥ 1, for all N ≥ N , N X j =1 ( j s − ( j − s ) (cid:16) N X µ = j x µ µ s (cid:17) ≤ C s N X µ =1 x µ µ. The conclusion thus follows from the afore mentionned Lubinsky’s result. (cid:3) Let s ≥ r > 0. Consider the more general integrals Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) r d tπ ( t + 1) , and in particular, for any positive integer k , I k ( N, σ, s ) = Z R (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) k d tπ ( t + 1) , corresponding to Dirichlet approximating polynomials. By simple iteration, Lemmas 1.1 and 1.3extend to general integer moments. Lemma 1.6. For any positive integer q , we have Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n − ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) = X ≤ µ ,...,µq ≤ N ≤ ν ,...,νq ≤ N x µ . . . x µ q x ν . . . x ν q (cid:16) ν . . . ν q ∧ µ . . . µ q ν . . . ν q ∨ µ . . . µ q (cid:17) s . And Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n − ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) = E (cid:12)(cid:12)(cid:12) X ≤ νi ≤ N ≤ i ≤ q x ν . . . x ν q W (( ν . . . ν q ) s )( ν . . . ν q ) s (cid:12)(cid:12)(cid:12) = N q X j =1 ( j s − ( j − s ) (cid:12)(cid:12)(cid:12) X ≤ νi ≤ N ≤ i ≤ qν ...νq ≥ j x ν . . . x ν q ( ν . . . ν q ) s (cid:12)(cid:12)(cid:12) . We omit the proof. By (3.1) and the considerations made after, it also follows that Corollary 1.7. Let q be a positive integer, s > and let F q ( t ) = (cid:0) P ∞ n =1 x n n − it (cid:1) q . F q ( s. ) ∈ L (cid:16) and thus Z R (cid:12)(cid:12)(cid:12) ∞ X n =1 x n n − ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) < ∞ (cid:17) if the Brownian sum X ≤ νi ≤ N ≤ i ≤ q x ν . . . x ν q W (( ν . . . ν q ) s )( ν . . . ν q ) s converges in L ( P ) . Of first importance in theprevious formulas is the role played by the parameter s , and more precisely the behavior of theCauchy means when s → ∞ .Lubinsky has established a clarifying link with mean-values of general Dirichlet polynomials.We state it under slightly weaker assumptions than in [3] p. 428. Lemma 1.8. Let g : R → C and define formally for any s ≥ , M ( s ) = s R s − s | g ( t ) | d t . Then, (1.10) Z +0 | g ( t ) | log 1 t d t < ∞ ⇐⇒ Z +0 M ( s ) d s < ∞ . (1.11) (cid:18) Z R | g ( t ) | t d t < ∞ and Z +0 | g ( t ) | log 1 t d t < ∞ (cid:19) ⇐⇒ Z R M ( s )1 + s d s < ∞ . MICHEL J. G. WEBER Under any of the previous properties, we further have (1.12) Z R | g ( st ) | d tπ (1 + t ) = 4 Z ∞ M ( su ) u π (1 + u ) d u. And if moreover, M ( s ) is locally bounded, then (1.13) lim s →∞ Z R | g ( st ) | d tπ (1 + t ) = lim s →∞ M ( s ) , if the preceding limit exists and is finite.Proof. Assertion (1.10) follows by integration by part. Further, if η > Z η ≤| t | < ∞ | g ( t ) | t d t < ∞ ⇐⇒ Z η ≤ s< ∞ M ( s )1 + s d s < ∞ . Hence (1.11) follows. An integration by part gives (1.12). Since M ( s ) → λ , say, and | λ | < ∞ ,there is a real A > Y > |M ( y ) | ≤ A if y ≥ Y . By assumption, M ( s ) is locally bounded, we also have M ( y ) ≤ B if 0 ≤ y ≤ Y . Thus M ( y ) ≤ A ∨ B on R + .Therefore M ( su ) u (1 + u ) ≤ ( A ∨ B ) u (1 + u ) ∈ L ( R + ) . And (1.13) follows from the dominated convergence theorem. (cid:3) Letting g = (cid:12)(cid:12) P ∞ n =1 x n n − it (cid:12)(cid:12) q , where q is a positive integer yieldslim s →∞ Z R (cid:12)(cid:12)(cid:12) ∞ X n =1 x n n − ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) = lim s →∞ s Z s − s (cid:12)(cid:12)(cid:12) ∞ X n =1 x n n − it (cid:12)(cid:12)(cid:12) q d t, provided that the second limit exists.Another link with standard mean-values of Dirichlet sums is provided with the next lemma. Lemma 1.9. Let q, S, T be positive reals. Then Z √ S + T S (cid:18) Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) (cid:19) d s = 12 π Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q log (cid:16) T θ + S (cid:17) d θ. Moreover, S Z S (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q d θ ≤ (cid:16) π log 2 (cid:17) sup S ≤ s ≤ S Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) . This provides a partial converse to Lubinsky’s observation. Indeed, assume thatlim sup s →∞ Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) = λ. Then it follows from the second part of the Lemma thatlim S →∞ S Z S (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q d θ ≤ (cid:16) π log 2 (cid:17) λ. Proof. By using the variable change t = θ/s , we get Z √ S + T S (cid:18) Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) (cid:19) d s = Z √ S + T S (cid:18) Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q s d θπ ( θ + s ) (cid:19) d s = Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q (cid:18) Z √ S + T S s d sπ ( θ + s ) (cid:19) d θ AUCHY MEANS OF DIRICHLET POLYNOMIALS 9 = 12 π Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q log (cid:16) T θ + S (cid:17) d θ. Letting T = √ S gives Z SS (cid:18) Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) (cid:19) d s = 12 π Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q log (cid:16) S θ + S (cid:17) d θ ≥ log 22 π Z S (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q d θ. Therefore Z S (cid:12)(cid:12)(cid:12) N X n =1 x n n iθ (cid:12)(cid:12)(cid:12) q d θ ≤ π log 2 Z SS (cid:18) Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) (cid:19) d s ≤ (cid:16) πS log 2 (cid:17) sup S ≤ s ≤ S Z R (cid:12)(cid:12)(cid:12) N X n =1 x n n ist (cid:12)(cid:12)(cid:12) q d tπ ( t + 1) . (cid:3) Finally, a simple re-summation argument also provides a direct connection with standard mean-values of Dirichlet polynomials. Lemma 1.10. There exist two positive absolute constants c, C such that c ∞ X j =1 M j j ≤ Z R (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) r d tπ ( t + 1) ≤ C ∞ X j =1 M j j . where we set M j = 12 j Z j − j (cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12) r d t, j = 1 , , . . . Remark . It is well-known that for any complex numbers x , . . . , x N and any 0 < α < ∞ , thelimits lim T →∞ T Z T (cid:12)(cid:12) N X n =1 x n n − it (cid:12)(cid:12) α d t exist. The series P ∞ j =1 M j j is thus convergent. For the values α = 2 k , k = 1 , , . . . , we recall thatlim T →∞ T Z T − T (cid:12)(cid:12)(cid:12) N X n =1 n σ + it (cid:12)(cid:12)(cid:12) k d t = X ≤ m ≤ N k d k,N ( m ) m σ , where d k,N ( m ) denotes the number of representations of m as a product of k factors less or equalto N . Proof of Lemma 1.10. Let u k = Z kk − (cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12) k d t, k = 1 , , . . . and note that ∞ X k =0 u k π ( k + 1) ≤ Z ∞ (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) k d tπ ( t + 1) ≤ ∞ X k =0 u k π (( k − + 1) , Let D j = P jk =1 u k , j ≥ 1. By applying Abel summation r X k =1 u k y k = D r y r +1 + r X j =1 D j ( y j − y j +1 ) , with y k = π (( k − +1) , we get Z r (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) k d tπ ( t + 1) ≤ π ( r + 1) Z r (cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12) k d t + r X j =1 j − π (( j − + 1)( j + 1) Z j (cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12) k d t. Hence Z ∞ (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) k d tπ ( t + 1) ≤ C ∞ X j =1 j Z j (cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12) k d t. Operating similarly for the lower part and next for the integration over R − provides the claimedestimate. (cid:3) In the next subsection, we investigate the behavior of Cauchy integrals when the parameter s is small and the moments are high. I k ( N, σ, s ) for s = s ( k ) small and k large. We now consider the behaviorof these integrals when s and k are simultaneously varying. More precisely, we will study the casewhen s = 1 / p c σ,N k where c σ,N ∼ c as k → ∞ ( c = c ( σ ) will be an explicit positive constant).We obtain the following very precise uniform estimate. Theorem 1.12. There exist two positive numerical constants c , C such that for all positiveintegers N , k and ≤ σ < , (cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) N X n =1 n σ + it/ √ c σ,N k (cid:12)(cid:12) k d tπ ( t + 1) − c (cid:0) N X n =1 n σ (cid:1) k (cid:12)(cid:12)(cid:12) ≤ C (1 − σ ) log Nk / (cid:16) N X n =1 n σ (cid:17) k where c σ,N = 2(1 − σ ) + O ( N σ − (log N ) ) . 2. Proof of Theorem 1.12 Our proof is probabilistic. We introduce a random model and first establish an interestingproperty (Lemma 2.3) of this one. We don’t know whether this model has been investigatedsomewhere. Let σ ≥ 0. Let N be some positive integer and note L N = P Nn =1 1 n σ .Let Y be random variable defined by(2.1) P { Y = log n } = 1 n σ L N , n = 1 , . . . , N. Let Y , . . . , Y k be independent copies of Y and note S k = Y + . . . + Y k . Lemma 2.1. Let e S k denote a symmetrization of S k . Then, (cid:12)(cid:12)(cid:12) N X n =1 n σ + it (cid:12)(cid:12)(cid:12) k = (cid:16) N X n =1 n σ (cid:17) k E e it e S k . Proof. We indeed have P { S k = log m } = X ≤ n ,...,nk ≤ Nn ...nk = m P (cid:8) Y = log n , . . . , Y k = log n k (cid:9) = δ k,N ( m ) m σ L kN where we set δ k,N ( m ) = (cid:8) ( n , . . . , n k ) ∈ { , N } k : m = n . . . n k (cid:9) .Further (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) k = (cid:16) N X m =1 m σ + ist (cid:17) k (cid:16) N X n =1 n σ − ist (cid:17) k = (cid:16) N k X µ =1 δ k,N ( µ ) µ σ + ist (cid:17)(cid:16) N k X ν =1 δ k,N ( ν ) ν σ − ist (cid:17) AUCHY MEANS OF DIRICHLET POLYNOMIALS 11 = L kN (cid:16) N k X µ =1 P { S k = log µ } µ ist (cid:17)(cid:16) N k X ν =1 P { S k = log ν } ν − ist (cid:17) = L kN (cid:12)(cid:12)(cid:12) N k X µ =1 P { S k = log µ } µ ist (cid:12)(cid:12)(cid:12) = L kN (cid:12)(cid:12) E e − istS k (cid:12)(cid:12) = L kN E e ist e S k . Hence, (cid:12)(cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12)(cid:12) k = (cid:16) N X n =1 n σ (cid:17) k E e ist e S k . (cid:3) Lemma 2.2. We have the relations T Z T − T (cid:12)(cid:12) N X n =1 n σ + it (cid:12)(cid:12) k d t = (cid:0) N X n =1 n σ (cid:1) k E sin T e S k T e S k Z R (cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12) k d tπ ( t + 1) = (cid:0) N X n =1 n σ (cid:1) k E e − s | e S k | . Proof. By Fubini’s theorem,1 T Z T − T (cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12) k d t = (cid:0) N X n =1 n σ (cid:1) k T E Z T − T e ist e S k d t = (cid:0) N X n =1 n σ (cid:1) k E sin sT e S k sT e S k . It also follows by integrating that Z R (cid:12)(cid:12) N X n =1 n σ + ist (cid:12)(cid:12) k d tπ ( t + 1) = (cid:0) N X n =1 n σ (cid:1) k Z R E e ist e S k d tπ ( t + 1)= (cid:0) N X n =1 n σ (cid:1) k E Z R e ist e S k d tπ ( t + 1)= (cid:0) N X n =1 n σ (cid:1) k E e − s | e S k | . A interesting fact of this model is that the variance of e Y is small (almost constant). This ismade precise in the lemma below. Lemma 2.3. Let < σ < . We have E e Y = 2 n N X m =1 (log m ) m σ L N − (cid:16) N X m =1 (log m ) m σ L N (cid:17) o = 2(1 − σ ) + O ( N σ − (log N ) ) . It will follow from the proof that the almost constant behavior of the variance arises fromcancellation of auxiliary sums. Proof. We use Euler-Maclaurin formula. Let h : [1 , N ] → R be a twice differentiable function.Then(2.2) N X k =1 h ( k ) = Z N h ( t ) dt + 12 ( h (1) + h ( N )) + N − X k =1 Z t − t h ′′ ( k + t ) dt. Applying this to h ( t ) = t α , − < α < 0, we get N X k =1 k α = N α +1 α + 1 + O (( N α )) + (cid:18) − α + 1 (cid:19) + α ( α − ∞ X k =1 Z t − t k + t ) α − dt − ∞ X k = N O (cid:0) k α − (cid:1) = N α +1 α + 1 + C α + O ( N α ) , where C α = 12 − α + 1 + α ( α − ∞ X k =1 Z t − t k + t ) α − dt . Thus L N = N − σ − σ + C − σ + O (cid:0) N − σ (cid:1) Apply it now to h ( t ) = (log t ) t − σ . We get N X k =1 log kk σ = N − σ log N − σ − N − σ − − σ ) + N − σ log N C ′ σ . Next N X k =1 (log k ) k σ = N − σ (log N ) − σ − N − σ (log N )(1 − σ ) + 2( N − σ − − σ ) + N − σ (log N ) C ′ σ . We moreover have 1 L N = 1 N − σ − σ + C σ + O ( N − σ ) . Therefore, N X k =1 (log k ) L N k σ = (cid:16) N − σ − σ + C σ (cid:17) N X k =1 (log k ) k σ + O ( N − (log N ) ) . Now (cid:16) N − σ − σ + C σ (cid:17) N X k =1 (log k ) k σ = (cid:16) N − σ − σ + C σ (cid:17)n N − σ (log N ) − σ − N − σ (log N )(1 − σ ) + 2( N − σ − − σ ) + N − σ (log N ) C ′ σ o = (cid:16) 11 + C σ N σ − (cid:17)n (log N ) − N − σ + 2(1 − σ ) − N σ − (1 − σ ) + N − (log N ) − σ ) C ′ σ N σ − o = (cid:16) 11 + C σ N σ − (cid:17)n (log N ) − N − σ + 2(1 − σ ) + O ( N σ − ) o . We have 1 − 11 + C σ N σ − = O ( N σ − ) . Therefore (cid:16) N − σ − σ + C σ (cid:17) N X k =1 (log k ) k σ = (cid:16) O ( N σ − ) (cid:17)n (log N ) − N − σ + 2(1 − σ ) + O ( N σ − ) o AUCHY MEANS OF DIRICHLET POLYNOMIALS 13 = (log N ) − N − σ + 2(1 − σ ) + O ( N σ − (log N ) ) + O ( N σ − )= (log N ) − N − σ + 2(1 − σ ) + O ( N σ − (log N ) ) . By reporting we get N X k =1 (log k ) L N k σ = (log N ) − N − σ + 2(1 − σ ) + O ( N σ − (log N ) ) . Similarly, N X k =1 log kL N k σ = (cid:16) N − σ − σ + C σ (cid:17) N X k =1 log kk σ + O ( N − log N )= (cid:16) N − σ − σ + C σ (cid:17)n N − σ log N − σ − N − σ − − σ ) + N − σ log N C ′ σ o + O ( N − log N ) . Further (cid:16) N − σ − σ + C σ (cid:17)n N − σ log N − σ − N − σ − − σ ) + N − σ log N C ′ σ o = (cid:16) 11 + C σ (1 − σ ) N σ − (cid:17)n log N − − σ + N σ − − σ + N − log N C ′ σ N σ − (1 − σ ) o = (cid:16) 11 + C σ (1 − σ ) N σ − (cid:17)n log N − − σ + O ( N σ − ) o = (cid:16) O ( N σ − ) (cid:17)n log N − − σ + O ( N σ − ) o = log N − − σ + O ( N σ − log N ) . Consequently, 12 E ( e Y ) = N X m =1 (log m ) m σ L N − (cid:16) N X m =1 log mm σ L N (cid:17) = (log N ) − N − σ + 2(1 − σ ) + O ( N σ − (log N ) ) − (cid:16) log N − − σ + O ( N σ − log N ) (cid:17) = 1(1 − σ ) + O ( N σ − (log N ) ) . (cid:3) It follows from the previous Lemma that s k = E ( e S k ) = k E ( e Y ) = 2 k (1 − σ ) + O ( kN σ − (log N ) ) . Choose s = 1 /s k . Let g be a Gaussian standard random variable. Then, Z R (cid:12)(cid:12) N X n =1 n σ + it/s k (cid:12)(cid:12) k d tπ ( t + 1) = (cid:0) N X n =1 n σ (cid:1) k (cid:8) E e −| g | + E e −| e S k | /s k − E e −| g | (cid:9) . Hence (cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) N X n =1 n σ + it/s k (cid:12)(cid:12) k d tπ ( t + 1) − (cid:0) N X n =1 n σ (cid:1) k E e −| g | (cid:12)(cid:12)(cid:12) = (cid:0) N X n =1 n σ (cid:1) k (cid:12)(cid:12) E e −| e S k | /s k − E e −| g | (cid:12)(cid:12) . By the transfert formula, (cid:12)(cid:12) E e −| e S k | /s k − E e −| g | (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z (cid:16) P (cid:8) e −| e S k | /s k > x (cid:9) − P (cid:8) e −| g | > x (cid:9)(cid:17) d x (cid:12)(cid:12)(cid:12) ( x = e − y ) = 2 (cid:12)(cid:12)(cid:12) Z ∞ (cid:16) P (cid:8) | e S k | /s k < y (cid:9) − P (cid:8) | g | < y (cid:9)(cid:17) ye − y d y (cid:12)(cid:12)(cid:12) ≤ x ∈ R (cid:12)(cid:12)(cid:12) P (cid:8) | e S k | /s k < x (cid:9) − P {| g | < x } (cid:12)(cid:12)(cid:12) ≤ A E | e Y | k / ( E | e Y | ) / , where we used Berry–Esseen theorem’s in the last inequality, A being a universal constant. Usingthe plain bound E | e Y | ≤ (log N ) E | e Y | , we therefore deduce (cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) N X n =1 n σ + it/s k (cid:12)(cid:12) k d tπ ( t + 1) − (cid:0) N X n =1 n σ (cid:1) k E e −| g | (cid:12)(cid:12)(cid:12) ≤ A log Nk / ( E | e Y | ) / (cid:0) N X n =1 n σ (cid:1) k ≤ C (1 − σ ) log Nk / (cid:16) N X n =1 n σ (cid:17) k . And C is a universal constant. By taking c = E e −| g | , this achieves the proof. (cid:3) 3. Concluding Remarks. The questions treated in [8] are also considered in [7] in the setting of Widom’s theory ofToeplitz integral kernels and their connection with finite sections of classical inequalities, such asCarleman or Hilbert’s inequality.We believe that these are really interesting and motivating questions, which should deservemore investigations, notably because of the connection with Dirichlet sums and the link withother approaches. We conclude with a simple remark concerning a second application of (1.1)(using the Hilbert kernel H ( x, y ) = ( x + y ) − ) given in [8], where the following formula in which σ > / λ ( n ) is the Liouville function is established,(3.1) ∞ X m =1 λ ( m ) m s X d | m d it d + ( m/d ) = Z ∞ (cid:12)(cid:12)(cid:12) ζ (cid:0) σ + ) + 2 i ( t + θ ) (cid:1) ζ (cid:0) σ + + i ( t + θ ) (cid:1) (cid:12)(cid:12)(cid:12) d θ cosh πθ . In fact, the same arguments used to establish (3.1) also apply for the kernel K ( x, y ) = max( x, y ) − ,and to other arithmetical functions. More precisely, let f ( n ) be a completely multiplicative arith-metical function. Assume that the series(3.2) ∞ X n =1 | f ( m ) | m σ converges for some σ > 1. Let F ( z ) = P ∞ m =1 f ( m ) m z . Then for σ ≥ σ − , (recalling that s = σ + it ),(3.3) ∞ X m =1 f ( m ) m s (cid:16) X d | m d it max (cid:0) d, ( m/d ) (cid:1) (cid:17) = 12 π Z R (cid:12)(cid:12) F (cid:0) σ + 12 + i ( t + θ ) (cid:1)(cid:12)(cid:12) d θ + θ . Indeed, by (1.1), N X n,m =1 f ( m ) f ( n ) m ¯ s n s K ( m, n ) = 12 π Z R (cid:12)(cid:12)(cid:12) N X n =1 f ( n ) n σ + + i ( t + θ ) (cid:12)(cid:12)(cid:12) d θ + θ . AUCHY MEANS OF DIRICHLET POLYNOMIALS 15 Now(3.4) N X n,m =1 f ( m ) f ( n ) m ¯ s n s K ( m, n ) = N X ν =1 f ( ν ) ν s (cid:16) X d | ννN ≤ d ≤ N K ( d, νd ) d it (cid:17) Further (cid:12)(cid:12)(cid:12) X d | ννN ≤ d ≤ N K ( d, νd ) d it (cid:12)(cid:12)(cid:12) ≤ X d | ννN ≤ d ≤ N d, ν/d ) = 1 √ ν X d | ννN ≤ d ≤ N d/ √ ν, √ ν/d ) ≤ d ( ν ) √ ν , where d ( n ) is the divisor function (counting the number of divisors of the natural n ), and we recallthat d ( n ) = O ε ( n ε ). Hence by assumption (3.2), (3.4) and letting N tend to infinity, the resultfollows. References [1] A. 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