(1)Problem 11739 (American Mathematical Monthly, Vol.120, November 2013) Proposed by Fred Adams, Anthony Bloch, and Jeffrey Lagarias (USA)

Problem 11739

(American Mathematical Monthly, Vol.120, November 2013) Proposed by Fred Adams, Anthony Bloch, and Jeffrey Lagarias (USA).

Let B(x) =

 1 x x 1



. Consider the infinite matrix product

M (t) =

∞

Y

n=1

B(p^−t_n ),

where pn is the nth prime. Evaluate M (2).

Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.

For N ≥ 0, let

MN(2) =

N

Y

n=1

B(p⁻²_n ) =

 a(N ) b(N ) b(N ) a(N )

 .

Then a(0) = 1, b(0) = 0 and

a(N ) = a(N − 1) +b(N − 1)

p²_N and b(N ) = b(N − 1) + a(N − 1) p²_N . Hence,

a(N ) + b(N ) = (a(N − 1) + b(N − 1))

 1 + 1

p²_N



=

N

Y

1

 1 + 1

p²_n



=

N

Y

1

 1 − 1

p⁴_n





 1 − 1

p²_n

N →∞

−→ ζ(2) ζ(4) = 15

π², and

a(N ) − b(N ) = (a(N − 1) − b(N − 1))

 1 − 1

p²_N



=

N

Y

1

 1 − 1

p²_n

N →∞

−→ 1

ζ(2) = 6 π².

Finally, M (2) =

 a b b a

 with

a = lim

N →∞a(N ) =1 2

 15 π² + 6

π²



= 21

2π², and b = lim

N →∞b(N ) = 1 2

 15 π² − 6

π²



= 9 2π².