Calculating a determinant associated with multiplicative functions

P. Codecá; M. Nair

Calculating a determinant associated with multiplicative functions

P. Codecá; M. Nair

Bollettino dell'Unione Matematica Italiana (2002)

Volume: 5-B, Issue: 2, page 545-555
ISSN: 0392-4041

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Abstract

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Let

h

be a complex valued multiplicative function. For any

N \in N

, we compute the value of the determinant

D_{N} := {det}_{i | N, j | N} (\frac{h ((i, j))}{i j})

where

(i, j)

denotes the greatest common divisor of

i

and

j

, which appear in increasing order in rows and columns. Precisely we prove that

D_{N} = \prod_{p^{l} ∥ N} {(\frac{1}{p^{l (l + 1)}} \overset{l}{\prod_{i = 1}} (h (p^{i}) - h (p^{i - 1})))}^{τ (N / p^{l})} .

This means that

D_{N}^{1 / τ (N)}

is a multiplicative function of

N

. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions

f (n)

, with

0 \leq f (p) < 1

, as minimal values of certain quadratic forms on the

τ (N)

unit sphere. The second one is the explicit evaluation of the minimal values of certain others quadratic forms also on the unit sphere.

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MLA
BibTeX
RIS

Codecá, P., and Nair, M.. "Calculating a determinant associated with multiplicative functions." Bollettino dell'Unione Matematica Italiana 5-B.2 (2002): 545-555. <http://eudml.org/doc/194699>.

@article{Codecá2002,
abstract = {Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb\{N\}$, we compute the value of the determinant $D_\{N\}:= \det _\{i|N, j|N\}\left(\frac\{h((i,j))\}\{ij\} \right)$ where $(i, j)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $$D\_\{N\}= \prod \_\{p^\{l\}\| N\}\left(\frac\{1\}\{p^\{l(l+1)\}\}\prod\_\{i=1\}^\{l\}(h(p^\{i\})-h(p^\{i-1\})) \right)^\{\tau (N/p^\{l\})\}.$$ This means that $D_\{N\}^\{1/\tau(N)\}$ is a multiplicative function of $N$. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions $f(n)$, with $0\leq f (p)<1$, as minimal values of certain quadratic forms on the $\tau(N)$ unit sphere. The second one is the explicit evaluation of the minimal values of certain others quadratic forms also on the unit sphere.},
author = {Codecá, P., Nair, M.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {545-555},
publisher = {Unione Matematica Italiana},
title = {Calculating a determinant associated with multiplicative functions},
url = {http://eudml.org/doc/194699},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Codecá, P.
AU - Nair, M.
TI - Calculating a determinant associated with multiplicative functions
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/6//
PB - Unione Matematica Italiana
VL - 5-B
IS - 2
SP - 545
EP - 555
AB - Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb{N}$, we compute the value of the determinant $D_{N}:= \det _{i|N, j|N}\left(\frac{h((i,j))}{ij} \right)$ where $(i, j)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $$D_{N}= \prod _{p^{l}\| N}\left(\frac{1}{p^{l(l+1)}}\prod_{i=1}^{l}(h(p^{i})-h(p^{i-1})) \right)^{\tau (N/p^{l})}.$$ This means that $D_{N}^{1/\tau(N)}$ is a multiplicative function of $N$. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions $f(n)$, with $0\leq f (p)<1$, as minimal values of certain quadratic forms on the $\tau(N)$ unit sphere. The second one is the explicit evaluation of the minimal values of certain others quadratic forms also on the unit sphere.
LA - eng
UR - http://eudml.org/doc/194699
ER -

References

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BELLMAN, R., Introduction to Matrix Analysis, McGraw-Hill, New York (1970). Zbl0216.06101 MR258847
CODECÁ, P.- NAIR, M., Extremal Values of $Δ (x, N) = \sum_{\begin{matrix} n \leq x N \\ (n, N) = 1 \end{matrix}} 1 - x ϕ (N)$ , Canad. Math. Bull. (3), 41 (1998), 335-347. Zbl0920.11066 MR1637673
DE BARTOLOMEIS, P., Algebra lineare, La Nuova Italia Editrice, Scandicci, Firenze1993.
GRAYBILL, F. A., Introduction to matrices with applications in statistics, Wordsworth Publishing CompanyBelmont, California1969. MR249443
PERELLI, A.- ZANNIER, U., An Extremal Property of the Möbius Function, Arch. Math., 53 (1989), 20-29. Zbl0683.10036 MR1005165

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