The proof of (49) rests on the realization that – because of symmetry – has only independent components and they are simply powers of and , so that may be written as with . Accordingly may be written as with .

n | N | n | N | ||

4 | 1 | 0.77459667 | 2600 | 23 | 6.59011627 |

10 | 2 | 1.34164079 | 2925 | 24 | 6.75262213 |

20 | 3 | 1.80822948 | 3276 | 25 | 6.91176615 |

35 | 4 | 2.21299946 | 3654 | 26 | 7.06774631 |

56 | 5 | 2.57495874 | 4060 | 27 | 7.22074198 |

84 | 6 | 2.90507811 | 4495 | 28 | 7.37091629 |

120 | 7 | 3.21035245 | 4960 | 29 | 7.51841807 |

165 | 8 | 3.49555791 | 5456 | 30 | 7.66338362 |

220 | 9 | 3.76412372 | 5984 | 31 | 7.80593804 |

286 | 10 | 4.01860847 | 6545 | 32 | 7.94619654 |

364 | 11 | 4.26098014 | 7140 | 33 | 8.08426549 |

455 | 12 | 4.26098014 | 7770 | 34 | 8.22024331 |

560 | 13 | 4.71528716 | 8436 | 35 | 8.35422129 |

680 | 14 | 4.92949284 | 9139 | 36 | 8.48628432 |

816 | 15 | 5.13625617 | 9880 | 37 | 8.61651144 |

969 | 16 | 5.33629130 | 10660 | 38 | 8.74497644 |

1140 | 17 | 5.53020569 | 11480 | 39 | 8.87174833 |

1330 | 18 | 5.71852112 | 12341 | 40 | 8.99689171 |

1540 | 19 | 5.90168962 | 13244 | 41 | 9.12046722 |

1771 | 20 | 6.08010585 | 14190 | 42 | 9.24253184 |

2024 | 21 | 6.25411673 | 15180 | 43 | 9.36313918 |

2300 | 22 | 6.42402919 | |||

Therefore (48) assumes the form

The elements of a semi-definite matrix satisfy the inequalities and therefore (50) implies

Since is an even function of we obtain

This estimate depends on the choice of the exponents through and we choose, rather arbitrarily , and all others zero. Also we set . In that case (52) implies which proves (49).An easy check will show that for each N the value lies below the corresponding values of Table 1, as they must. It may well be possible to tighten the estimate (49).

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