A MATRIX INEQUALITY FOR MÖBIUS FUNCTIONS OLIVIER BORDELLÈS
BENOIT CLOITRE
2 allée de la combe 43000 AIGUILHE (France) EMail:
[email protected]
19 rue Louise Michel 92300 LEVALLOIS-PERRET (France) EMail:
[email protected]
Möbius Functions Olivier Bordellès and Benoit Cloitre vol. 10, iss. 3, art. 62, 2009
Title Page Received:
24 November, 2008
Accepted:
27 March, 2009
Communicated by:
L. Tóth
2000 AMS Sub. Class.:
15A15, 11A25, 15A18, 11C20.
Key words:
Determinants, Dirichlet convolution, Möbius functions, Singular values.
Abstract:
The aim of this note is the study of an integer matrix whose determinant is related to the Möbius function. We derive a number-theoretic inequality involving sums of a certain class of Möbius functions and obtain a sufficient condition for the Riemann hypothesis depending on an integer triangular matrix. We also provide an alternative proof of Redheffer’s theorem based upon a LU decomposition of the Redheffer’s matrix.
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Contents 1
Introduction 1.1 Arithmetic motivation . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Convolution identities for the Möbius function . . . . . . . . . . . .
2
An Integer Matrix Related to the Möbius Function 2.1 The determinant of Γn . . . . . . . . . . . . . . . . . 2.2 A sufficient condition for the PNT and the RH . . . . . 2.2.1 Computation of Un−1 . . . . . . . . . . . . . . 2.2.2 A sufficient condition for the PNT and the RH
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3 3 4 7 7 10 10 15
Möbius Functions Olivier Bordellès and Benoit Cloitre vol. 10, iss. 3, art. 62, 2009
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1.
Introduction
In what follows, [t] is the integer part of t and, for integers i, j > 1, we set mod (j, i) := j − i[j/i]. 1.1.
Arithmetic motivation
In 1977, Redheffer [5] introduced the matrix Rn = (rij ) ∈ Mn ({0, 1}) defined by ( 1, if i | j or j = 1; rij = 0, otherwise
Olivier Bordellès and Benoit Cloitre vol. 10, iss. 3, art. 62, 2009
Title Page
and has shown that (see appendix) det Rn = M (n) :=
Möbius Functions
n X
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µ(k),
k=1
where µ is the Möbius function and M is the Mertens function. This determinant is clearly related to two of the most famous problems in number theory, the Prime Number Theorem (PNT) and the Riemann Hypothesis (RH). Indeed, it is well-known that PNT ⇐⇒ M (n) = o(n) and RH ⇐⇒ M (n) = Oε n1/2+ε (for any ε > 0). These estimations for | det Rn | remain unproven, but [6] h Vaughan i log n showed that 1 is an eigenvalue of Rn with (algebraic) multiplicity n− log 2 −1, that Rn has two "dominant" eigenvalues λ± such that |λ± | n1/2 , and that the others eigenvalues satisfy λ (log n)2/5 .
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It should be mentioned that Hadamard’s inequality, which states that n Y | det Rn | 6 kLi k22 , 2
i=1
where Li is the ith row of Rn and k·k2 is the euclidean norm on Cn , gives 2
(M (n)) 6 n
n Y i=2
1+
h n i i
n−[n/2]
=2
n
[n/2]
Y i=2
1+
h n i i
n−[n/2]
62
n + [n/2] , n
which is very far from the trivial bound |M (n)| 6 n so that it seems likely that general matrix analysis tools cannot be used to provide an elementary proof of the PNT. In this work we study an integer matrix whose determinant is also related to the Möbius function. This will provide a new criteria for the PNT and the RH (see Corollary 2.3 below). In an attempt to go further, we will prove an inequality for a class of Möbius functions and deduce a sufficient condition for the PNT and the RH in terms of the smallest singular value of a triangular matrix.
Möbius Functions Olivier Bordellès and Benoit Cloitre vol. 10, iss. 3, art. 62, 2009
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1.2.
Convolution identities for the Möbius function
The function µ, which plays an important role in number theory, satisfies the following well-known convolution identity. Lemma 1.1. For every real number x > 1 we have hxi X x X µ(k) = M = 1. k d k6x d6x One can find a proof for example in [1]. The following corollary will be useful.
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Corollary 1.2. For every integer j > 1 we have (i) j X µ(k) mod (j, k)
k
k=1
=j
j X µ(k) k=1
k
− 1.
(ii) j k X X µ(h) h k=1 h=1
Möbius Functions
! (mod(j, k + 1) − mod(j, k)) = 1.
Olivier Bordellès and Benoit Cloitre vol. 10, iss. 3, art. 62, 2009
Proof. Title Page
(i) We have j X µ(k) mod (j, k) k=1
k
=
j X µ(k) k=1
k
j j X µ(k) X j j j−k =j − µ(k) k k k k=1 k=1
and we conclude with Lemma 1.1.
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(ii) Using Abel summation we get ! j k X X µ(h) (mod(j, k + 1) − mod(j, k)) h k=1 h=1 ! j j X µ(h) X (mod(j, k + 1) − mod(j, k)) = h h=1 k=1 ! k j−1 k+1 k X X µ(h) X µ(h) X − − (mod(j, m + 1) − mod(j, m)) h h m=1 k=1 h=1 h=1
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=j
j X µ(k) k=1 j
=j
k
X µ(k) k=1
k
−
j−1 X µ(k + 1) mod (j, k + 1)
k+1
k=1 j
−
X µ(k) mod (j, k) k=1
and we conclude using (i).
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2.
An Integer Matrix Related to the Möbius Function
We now consider the matrix Γn = (γij ) defined by mod(j, 2) − 1, if i = 1 and 2 6 j 6 n; mod(j, i + 1) − mod(j, i), if 2 6 i 6 n − 1 and 1 6 j 6 n; γij = 1, if (i, j) ∈ {(1, 1), (n, 1)}; 0, otherwise. The matrix Γn is almost upper triangular except the entry γn1 = 1 which is nonzero. Note that it is easy to check that |γij | 6 i for every 1 6 i, j 6 n and that γij = −1 if [j/2] < i < j. Example 2.1. 1 −1 0 −1 0 −1 0 −1 0 2 −1 1 1 0 0 2 3 −1 −1 2 2 −2 0 0 0 4 −1 −1 −1 3 0 0 Γ8 = . 0 0 5 −1 −1 −1 0 0 0 0 0 0 0 6 −1 −1 0 0 0 0 0 0 7 −1 1 0 0 0 0 0 0 0 2.1.
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The determinant of Γn
Theorem 2.1. Let n > 2 be an integer and Γn defined as above. Then we have det Γn = n!
n X µ(k) k=1
k
.
A possible proof of Theorem 2.1 uses a LU decomposition of the matrix Γn . Let Ln = (lij ) and Un = (uij ) be the matrices defined by 0, if (i, j) = (n, 1); uij = 1, if (i, j) = (n, n); γij , otherwise
Möbius Functions Olivier Bordellès and Benoit Cloitre
and
lij =
1, Pj
k=1
if 1 6 i = j 6 n − 1; µ(k) , k
Pn n k=1 0,
µ(k) , k
if i = n
and 1 6 j 6 n − 1;
otherwise.
Lemma 2.2. We have Γn = Ln Un .
Moreover, using Corollary 1.2 (ii) we get for i = n and 2 6 j 6 n − 1 lnk ukj = ln1 u1j +
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k=1
k=1
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Proof. Set Ln Un = (xij ). When i = 1 we immediately obtain x1j = u1j = γ1j . We also have n X xn1 = lnk uk1 = ln1 u11 = 1 = γn1 .
xnj =
Title Page
if (i, j) = (n, n);
The proof of Theorem 2.1 follows from the lemma below.
n X
vol. 10, iss. 3, art. 62, 2009
n X k=2
lnk ukj
! j k X X µ(h) = mod(j, 2) − 1 + (mod(j, k + 1) − mod(j, k)) h k=2 h=1 ! j k X X µ(h) = (mod(j, k + 1) − mod(j, k)) − 1 = 0 = γnj h k=1 h=1 and, for (i, j) = (n, n), we have similarly xnn =
n X
lnk ukn = ln1 u1n +
k=1
= mod(n, 2) − 1 +
n−1 X
Möbius Functions Olivier Bordellès and Benoit Cloitre
lnk ukn + lnn unn
n−1 X
k=2 k X
k=2
h=1
µ(h) h
! (mod(n, k + 1) − mod(n, k))
n X
µ(k) k k=1 ! n k X X µ(h) = (mod(n, k + 1) − mod(n, k)) − 1 = 0 = γnn . h k=1 h=1 +n
Finally, for 2 6 i 6 n − 1 and 1 6 j 6 n, we get xij =
n X k=1
lik ukj = lii uij = uij = γij .
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Example 2.2. 1 0 0 1 0 0 0 0 Γ8 = 0 0 0 0 0 0 1 12
0 0 1 0 0 0 0
0 0 0 1 0 0 0
1 6
1 6
0 0 0 0 1 0 0 1 − 30
0 0 0 0 0 1 0 2 15
0 0 1 −1 0 −1 0 −1 0 −1 0 0 0 2 −1 1 1 0 0 2 0 0 0 0 3 −1 −1 2 2 −2 0 0 0 0 0 4 −1 −1 −1 3 . 0 0 0 0 0 0 5 −1 −1 −1 0 0 0 0 0 0 0 6 −1 −1 1 0 0 0 0 0 0 0 7 −1 1 8 0 0 0 0 0 0 0 1 − 105 − 105
Möbius Functions Olivier Bordellès and Benoit Cloitre vol. 10, iss. 3, art. 62, 2009
Theorem 2.1 now immediately follows from Title Page
det Γn = det Ln det Un = (n − 1)! det Ln = n!
n X µ(k) k=1
k
.
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We easily deduce the following criteria for the PNT and the RH. Corollary 2.3. For any real number ε > 0 we have PNT ⇐⇒ det Γn = o(n!) and 2.2.
RH ⇐⇒ det Γn = Oε (n
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n!).
A sufficient condition for the PNT and the RH
2.2.1. Computation of Un−1
The inverse of Un uses a Möbius-type function denoted by µi which we define below. Definition 2.4. Set µ1 = µ the well-known Möbius function and, for any integer
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i > 2, we define the Möbius function µi by µi (1) = 1 and, for any integer m > 2, by m µ i , if i | m and (i + 1) - m; m −µ i+1 , if (i + 1) | m and i - m; µi (m) := m µ mi − µ i+1 , if i(i + 1) | m; 0, otherwise. The following result completes and generalizes Lemma 1.1 and Corollary 1.2. Lemma 2.5. For all integers i, j > 2 we have j X j µi (k) = δij , k k=i j X µi (k) mod (j, k)
k
k=i j
k X X µi (h) k=i
h=i
h
=j
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j X µi (k) k=i
Möbius Functions
k
− δij ,
!
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(mod(j, k + 1) − mod(j, k)) = δij ,
where δij is the Kronecker symbol. Proof. We only prove the first identity, the proof of the two others being strictly identical to the identities of Corollary 1.2. Without loss of generality, one can suppose that 2 6 i 6 j. If i = j then we have j X j j j µi (k) = µj (j) =µ = 1. k j j k=i
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Now suppose that 2 6 i < j. By Lemma 1.1, we have j X
j µi (k) = k k=i
j X k=i i|k, (i+1)-k
k j µ − i k
j X
µ
k=i (i+1)|k, i-k
k i+1
j k
j X k k j + µ −µ i i+1 k k=i i(i+1)|k
j j X X k j k j = µ µ − i k i+1 k k=i k=i i|k
(i+1)|k
[j/i] X
Möbius Functions Olivier Bordellès and Benoit Cloitre vol. 10, iss. 3, art. 62, 2009
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[j/(i+1)] X [j/i] [j/(i + 1)] µ(h) = − µ(h) h h h=1 h=1
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This result gives the inverse of Un .
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Corollary 2.6. Set Un−1 = (θij ). Then we have θij =
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j X µi (k) k=i n X
θin = n
k=i
k µi (k) k
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(1 6 i 6 j 6 n − 1) (1 6 i 6 n).
Proof. Since Un−1 is upper triangular, it suffices to show that, for all integers 1 6 i 6 j 6 n, we have j X θik ukj = δij . k=i
In what follows, we set Sij as the sum on the left-hand side We easily check that Sjj = 1 for every integer 1 6 j 6 n. Now suppose that 1 6 i < j 6 n − 1. By Corollary 1.2, we first have S1j =
j X
θ1k ukj = θ11 u1j +
k=1
j X
Olivier Bordellès and Benoit Cloitre vol. 10, iss. 3, art. 62, 2009
θ1k ukj
k=2 j
= mod(j, 2) − 1 +
k X X µ(h) k=2
j k X X µ(h) = h k=1 h=1
Möbius Functions
h=1
h
!
Title Page
(mod(j, k + 1) − mod(j, k))
! (mod(j, k + 1) − mod(j, k)) − 1 = 0
and, if 2 6 i < j 6 n − 1, then by Lemma 2.5, we have ! j k X X µi (h) (mod(j, k + 1) − mod(j, k)) = δij = 0. Sij = h k=i h=1 Now suppose that j = n. By Corollary 1.2, we first have ! n n−1 k X X X µ(h) θ1k ukn = θ11 u1n + (mod(n, k + 1) − mod(n, k)) + θ1n unn S1n = h k=1 k=2 h=1 ! n−1 k n X X X µ(h) µ(k) = mod(n, 2) − 1 + (mod(n, k + 1) − mod(n, k)) + n h k k=2 h=1 k=1
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n k X X µ(h) = h k=1 h=1
! (mod(n, k + 1) − mod(n, k)) − 1 = 0
and, if 2 6 i 6 n − 1, we have ! n−1 k X X µi (h) Sin = (mod(n, k + 1) − mod(n, k)) + θin unn h k=i h=i ! n n−1 k X X X µi (h) µi (k) (mod(n, k + 1) − mod(n, k)) + n = h k k=i k=i h=i ! n k X X µi (h) = (mod(n, k + 1) − mod(n, k)) = δin = 0 h k=i h=i which completes the proof. For example, we get
U8−1
1 0 0 0 = 0 0 0 0
1 2 1 2
0 0
1 12 1 4
1 − 30 1 − 12 1 12 1 20
2 15 1 − 12 1 − 12 1 20
1 − 105 1 − 12 1 − 12 1 20
0 0
0
1 5
0 0
0
0
1 30 1 6
0 0
0
0
0
1 30 1 42 1 7
4 15 4 21 1 7
0 0
0
0
0
0
1
0
1 6 1 6 1 3
1 6 1 − 12
8 − 105 − 23 1 3 − 35
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2.2.2. A sufficient condition for the PNT and the RH
Corollary 2.7. For all integers i > 1 and n > 2 we have n X µi (k) 1 , 6 k nσn k=i where σn is the smallest singular value of Un . Thus any estimate of the form σn ε n−1+ε , where ε > 0 is any real number, is sufficient to prove the PNT. Similarly, any estimate of the form σn ε n−1/2−ε where ε > 0 is any real number, is sufficient to prove the RH. Proof. The result follows at once from the well-known inequalities kUn−1 k2
> max |θij | > |θin | 16i,j6n
(see [3]), where k·k2 is the spectral norm, and the fact that σn = kUn−1 k−1 2 . Smallest singular values of triangular matrices have been studied by many authors. For example (see [2, 4]), it is known that, if An = (aij ) is an invertible upper triangular matrix such that |aii | > |aij | for i < j, then we have σn >
min |aii | 2n−1
but, applied here, such a bound is still very far from the PNT.
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Appendix : A Proof of Redheffer’s Theorem Let Sn = (sij ) and Tn = (tij ) be the matrices defined by M (n/i), ( 1, if i | j; sij = and tij = 1, 0, otherwise 0,
if j = 1; if i = j > 2; otherwise.
Möbius Functions Olivier Bordellès and Benoit Cloitre
Proposition 2.8. We have Rn = Sn Tn . In particular, det Rn = M (n). Proof. Set Sn Tn = (xij ). If j = 1, by Lemma 1.1, we have n X X n X n/i xi1 = sik tk1 = M = M = 1 = ri1 . k d k=1 k6n d6n/i
i|k
If j > 2, then t1j = 0 and thus xij =
n X k=2
sik tkj = sij =
( 1, 0,
if i | j otherwise
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= rij
which is the desired result. The second assertion follows at once from
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det Rn = det Sn det Tn = det Tn = M (n). The proof is complete.
References [1] O. BORDELLÈS, Thèmes d’arithmétique, Editions Ellipses, 2006. [2] N.J. HIGHAM, A survey of condition number for triangular matrices, Soc. Ind. Appl. Math., 29 (1987), 575–596. [3] R.A. HORN Press, 1985.
AND
C.R. JOHNSON, Matrix Analysis, Cambridge University
Möbius Functions Olivier Bordellès and Benoit Cloitre
[4] F. LEMEIRE, Bounds for condition number of triangular and trapezoid matrices, BIT, 15 (1975), 58–64.
vol. 10, iss. 3, art. 62, 2009
[5] R.M. REDHEFFER, Eine explizit lösbare Optimierungsaufgabe, Internat. Schiftenreihe Numer. Math., 36 (1977), 213–216.
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[6] R.C. VAUGHAN, On the eigenvalues of Redheffer’s matrix I, in : Number Theory with an Emphasis on the Markoff Spectrum (Provo, Utah, 1991), 283–296, Lecture Notes in Pure and Appl. Math., 147, Dekker, New-York, 1993.
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