#A56 INTEGERS 20 (2020)

A NOTE ON A UNITARY ANALOG TO REDHEFFER’S

Olivier Bordell`es Aiguilhe, France [email protected]

Received: 11/2/19, Accepted: 7/7/20, Published: 7/24/20

Abstract We study a unitary analog to Redhe↵er’s matrix. It is first proved that the deter- minant of this matrix is the unitary analogue to that of Redhe↵er’s matrix. We also show that the coecients of the characteristic polynomial may be expressed as sums of Stirling numbers of the second kind. This implies in particular that 1 is an eigenvalue with algebraic multiplicity greater than that of Redhe↵er’s matrix.

1. Introduction

In 1977, Redhe↵er [7] introduced the matrix R = (r ) ( 0, 1 ) defined by n ij 2 Mn { } 1, if i j or j = 1 rij = | (0, otherwise and has shown that n det Rn = M(n) := µ(k), kX=1 where µ is the M¨obius function and M is the . This is clearly related to two of the most famous problems in number theory, namely the Prime Number Theorem (PNT) and the (RH) since it is well-known that

PNT M(n) = o(n) and RH M(n) = O n1/2+" . () () " ⇣ ⌘ The second estimate remains unproven, but Vaughan [11] showed that 1 is an eigen- value of R with algebraic multiplicity n log n 1, that R has two “domi- n log 2 n nant” eigenvalues such that n1/2, andj thatk the other eigenvalues satisfy ± | ±| ⇣ (log n)2/5. ⌧ INTEGERS: 20 (2020) 2

The purpose of this note is to supply an analogous study to the 0, 1 -matrix { } Rn⇤ = (⇢ij) defined by 1, if i j or j = 1 ⇢ij = k (0, otherwise. Recall that the integer i is said to be a unitary divisor of j, denoted by i j, k whenever i j and gcd i, j = 1. | i For instance, when n = 8, we have 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 01 0 1 0 0 1 0 01 B1 0 0 1 0 0 0 0C R⇤ = B C . 8 B1 0 0 0 1 0 0 0C B C B1 0 0 0 0 1 0 0C B C B1 0 0 0 0 0 1 0C B C B1 0 0 0 0 0 0 1C B C @ A Note that this matrix does not belong to the set of general matrices studied in [2].

This article is organized as follows. In Section 2, we shall use some elementary properties of unitary divisors to determine an LU-decomposition of the matrix Rn⇤ and deduce its determinant. In Section 3, following the ideas of [11], we shall discuss further on the characteristic polynomial of Rn⇤ and the algebraic multiplicity of the eigenvalue 1 of this matrix.

1.1. Notation

In what follows, n > 2 is a fixed integer and the function µ⇤ is the unitary analog of the M¨obius function. We also define

M ⇤(x, n) := µ⇤(k) (x > 0, n N) 2 k6x gcd(Xk,n)=1 and simply write M ⇤(x) := M ⇤(x, 1) which is the unitary analog of the Mertens function. As usual, let 1(n) = 1 and the unitary convolution product of the two arithmetic functions f and g is defined by

(f g)(n) = f(d)g(n/d). d n Xk Finally, from [3, Theorem 2.5] it is known that

!(n) µ⇤(n) = ( 1) , INTEGERS: 20 (2020) 3 where !(n) is the number of distinct prime factors of n, and from [3, Corollary 2.1.2] we have the important convolution identity

1, if n = 1 (µ⇤ 1) (n) = (1) (0, otherwise.

2. The Determinant of Rn⇤

We start with the following basic identities involving unitary divisors which will prove to be useful to determine an LU-type decomposition of the matrix Rn⇤ . Lemma 1. (i) Let i, j be positive integers. Then

1, if i = j µ⇤(d) = 0, otherwise. d j ( iXjk/d k (ii) Let 1 i n be integers. Then   n M ⇤ , k = 1. k k n Xik ⇣ ⌘ k Proof.

(i) If i , j, then the sum is equal to 0 since (d j and i j/d) implies i j. k k k If i j, then k (d j and i j/d) if and only if d j/i k k k so that using (1) we get

1, if j/i = 1 µ⇤(d) = µ⇤(d) = 0, otherwise. d j d j/i ( iXjk/d Xk k (ii) Using the identity above, we get j n 1 = µ⇤ = µ⇤(d) = M ⇤ , k . k k j n k j k n d n/k k n  ✓ ◆ ⇣ ⌘ X Xi kk Xik gcd(Xd,k)=1 Xik k k k The proof is complete. INTEGERS: 20 (2020) 4

Let S = (s ) and T = (t ) be the (n n)-matrices defined by n ij n ij ⇥

M ⇤(n/i, i), if j = 1 1, if i j sij = k and tij = 81, if i = j 2 (0, otherwise <>0, otherwise.

For instance :> 1 1 1 1 1 1 1 1 4 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 00 0 1 0 0 1 0 01 0 0 0 1 0 0 0 0 0 1 B0 0 0 1 0 0 0 0C B 1 0 0 1 0 0 0 0 C S = B C and T = B C . 8 B0 0 0 0 1 0 0 0C 8 B 1 0 0 0 1 0 0 0 C B C B C B0 0 0 0 0 1 0 0C B 1 0 0 0 0 1 0 0 C B C B C B0 0 0 0 0 0 1 0C B 1 0 0 0 0 0 1 0 C B C B C B0 0 0 0 0 0 0 1C B 1 0 0 0 0 0 0 1 C B C B C @ A @ A We are now in a position to prove the first result concerning the matrix Rn⇤ .

Theorem 2. Let n 2 be an integer. Then R⇤ = S T . In particular n n n n

det Rn⇤ = M ⇤(n) = µ⇤(k). kX=1

Proof. Set SnTn = (xij). If j = 1, using Lemma 1 (ii) we get

n n x = s t = M ⇤ , k = 1 = ⇢ . i1 ik k1 k i1 k=1 k6n X Xi k ⇣ ⌘ k

If j > 2, then t1j = 0 and thus

n 1, if i j xij = siktkj = sij = k = ⇢ij, (0, otherwise kX=2 which is the desired result. The second assertion follows at once from

det Rn⇤ = det Sn det Tn = det Tn = M ⇤(n).

The proof is complete.

Corollary 3. The Riemann hypothesis is true if and only if, for each " > 0

1/2+" det Rn⇤ = O n . ⇣ ⌘ INTEGERS: 20 (2020) 5

3. The Characteristic Polynomial of Rn⇤ 3.1. The “Trivial” Eigenvalue 1

log n Let ` = log 2 . It is proved in [11] that 1 is an eigenvalue of the Redhe↵er’s matrix Rn of algebraicj k multiplicity equal to n ` 1. We will show in this section that the algebraic multiplicity mn of the eigenvalue 1 of Rn⇤ may be somewhat larger. To this end, we first note that the method developed in [2, 11] to determine the characteristic polynomial of Redhe↵er type matrices can readily be adapted to the matrix Rn⇤ , which yields

` n n 2 n k 1 det (I R⇤ ) = ( 1) (n 1) ( 1) S⇤(n) ( 1) , n n k kX=2 where

Sk⇤(x) = Dk⇤(m) mX6x and

Dk⇤(m) = 1.

m=d1 dk i=j gcdX(d···,d )=1 6 ) i j dj >2

Note that the Dk⇤ is the unitary analogue to the strict divisor function Dk, which can be found in the coecients of the characteristic polynomial of Rn. Hence, using [10, (14)] and [1, (4)] successively, we get for any m, k Z 1 2 > k k k j k k j k !(m) !(m) D⇤(m) = ( 1) ⌧ ⇤(m) = ( 1) j = k! , k j j j k j=0 j=0 X ✓ ◆ X ✓ ◆ ⇢ where n is the Stirling number of the second kind. In particular, for any m, k k 2 such that !(m) < k, we have D (m) = 0. We are now in a position to prove Z>1 k⇤ the following result.

Theorem 4. Let n > 1. Then the algebraic multiplicity mn of the eigenvalue 1 of Rn⇤ satisfies m = n k , n n where the sequence (kn) of positive integers is given by

k1 = 0 and kn = max (kn 1, !(n) + 1) (n Z 2) . (2) 2 > In particular

1.3841 log n (n>3) (n>6) log n n 1 m n . log log n 6 n 6 log log n ⌫ ⌫ INTEGERS: 20 (2020) 6

Also, for any n > 3 log n 2 log n m = n + O? . n log log n (log log n)2 ✓ ◆ Proof. Since m = 1 = 1 k , we may suppose n 2. We first show by induction 1 1 > that, for any n Z 2, there exists a sequence (kn) of positive integers such that, 2 > for any m 1, . . . , n , !(m) < k , this sequence being given by (2). Indeed, 2 { } n the assertion is obviously true for n = 2 since k2 = 2, and if we assume it for some n 2, then, for any m 1, . . . , n + 1 , either m 1, . . . , n and then > 2 { } 2 { } !(m) < kn by induction hypothesis, or m = n + 1 and !(m) < 1 + !(n + 1), so that, for any m 1, . . . , n + 1 , we get !(m) < max (k , !(n + 1) + 1) = k . 2 { } n n+1 We now prove that kn is the smallest nonnegative integer satisfying this property, i.e., if there exists hn Z 0 such that, for all m 1, . . . , n , !(m) < hn, then 2 > 2 { } kn hn. Suppose on the contrary that hn < kn = max (kn 1, !(n) + 1). If 6 hn < !(n) + 1, then !(n) < hn < !(n) + 1 giving a contradiction, and hence hn < kn 1 = max (kn 2, !(n 1) + 1). Again, if hn < !(n 1)+1, then !(n 1) < hn < !(n 1) + 1 which is impossible, and hence hn < kn 2. Continuing this way we finally get hn < k1 = 1, resulting in a contradiction. Hence for any m 1, . . . , n , we infer that D⇤(m) = 0 for any k k , and thus 2 { } k > n S⇤(n) = 0 (k k ) and S⇤(n) = 0 (k < k ) , k > n k 6 n completing the proof of the first part of the theorem. For the second part, we first numerically check the inequality for n 3, . . . , 29 and assume n 30, so that 2 { } > kn 4. Next, for any k Z 1, define Nk := p1 pk. It is easy to see that kn is > 2 > · · · the unique positive integer such that Nk 1 n < Nk (see also [8, p. 380]), so n 6 n that, from [8, Theorem 11], we derive

1.3841 log Nk 1 1.3841 log n n kn = 1 + ! (Nkn 1) 6 1 + 6 1 + . log log Nk 1 log log n n Furthermore, [8, Theorem 10] yields

log Nkn log n kn = ! (Nkn ) > > , log log Nkn log log n which proves the inequality. We proceed similarly for the last estimate: first check it for n 3, . . . , 2 309 , then assume n 2 310 so that k 6, and use [8, 2 { } > n > Theorem 12] to get

log Nk 1 1.4575 log Nk 1 n n kn 6 1 + + 2 log log Nkn 1 (log log Nk 1) n log Nk 1 2 log Nk 1 n n < + 2 log log Nkn 1 (log log Nk 1) n log n 2 log n 6 + , log log n (log log n)2 INTEGERS: 20 (2020) 7 which terminates the proof of Theorem 4.

3.2. The “Dominant” Eigenvalues We first notice that

!(m) !(m) S⇤(x) = 2 = 2 2 x 2 2 b c mX6x ⇢ mX6x x log x 3 ⇣ = + 2x 0 (2) + o x1/2 . ⇣(2) 2 ⇣ ✓ ◆ ⇣ ⌘

Now following the argument leading to [11, (18)], we deduce that Rn⇤ has two “dominant” eigenvalues satisfying the following estimate. ±

Proposition 5. For all n Z 3 2 >

log n 1 ⇣ 0 1/2 2 = pn + + (2) + O n log n . ± ± 2⇣(2) 2 ⇣ ⇣ ⌘

Acknowledgments. The author gratefully acknowledges the anonymous referee for some corrections and remarks that have significantly improved the paper.

References

[1] K. N. Boyadzhiev, Close encounters with the Stirling numbers of the second kind, Math. Mag. 85 (2012), 252–266.

[2] D. A. Cardon, Matrices related to , J. Number Theory 130 (2010), 27–39.

[3] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Z. 74 (1960), 66–80.

[4] M. El Marraki, Fonction sommatoire de la fonction de M¨obius, 3. Majorations e↵ectives fortes, J. Th´eorie des Nombres de Bordeaux 7 (1995), 407–433.

[5] R. R. Hall and G. Tenenbaum, Divisors, Cambridge Tracts in Mathematics 90, Cambridge University Press, 1988.

[6] O. Ramar´e, Arithmetical Aspects of the Large Sieve Inequality, Vol. 1. Harish-Chandra Re- search Institute Lecture Notes. With the collaboration of D. S. Ramana. New Delhi: Hindustan Book Agency, 2009, pp. x+201.

[7] R. M. Redhe↵er, Eine explizit l¨osbare Optimierungsaufgabe, Internat. Schiftenreihe Numer. Math. 36 (1977), 213–216.

[8] G. Robin, Estimation de la fonction de Tchebychef ✓ sur le k-i`eme nombre premier et grandes valeurs de la fonction !(n) nombre de diviseurs de n, Acta Arith. 42 (1983), 367–389. INTEGERS: 20 (2020) 8

[9] J. B. Rosser and L. Schœnfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94.

[10] J. S´andor, On the arithmetical functions dk(n) and dk⇤(n), Portugaliæ Math. 53 (1996), 107–115.

[11] R. C. Vaughan, On the eigenvalues of Redhe↵er’s matrix I, in : Number theory with an emphasis on the Marko↵ spectrum (Provo, Utah, 1991), 283–296, Lecture notes in pure and Appl. Math 147, Dekker, New-York, 1993.