Bounded Dirichlet inverse: Difference between revisions
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New page: Every completely-multiplicative sequence has Dirichlet inverse bounded by <math>1</math>, and the longest discrepancy-2 sequence with this property has length <math>246</math>. (Are all su... |
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Every completely-multiplicative sequence has Dirichlet inverse bounded by <math>1</math> | Every completely-multiplicative <math>\pm 1</math> sequence has Dirichlet inverse bounded by <math>1</math> (express the Dirichlet series as a product over primes). The longest discrepancy-2 sequence with this property has length <math>246</math>. (Are all such maximal sequences completely multiplicative?) | ||
The longest discrepancy-2 sequence with Dirichlet inverse bounded by <math>2</math> has length <math>389</math>. Here is an example: | The longest discrepancy-2 sequence with Dirichlet inverse bounded by <math>2</math> has length <math>389</math>. Here is an example: | ||
Latest revision as of 22:03, 21 June 2010
Every completely-multiplicative [math]\displaystyle{ \pm 1 }[/math] sequence has Dirichlet inverse bounded by [math]\displaystyle{ 1 }[/math] (express the Dirichlet series as a product over primes). The longest discrepancy-2 sequence with this property has length [math]\displaystyle{ 246 }[/math]. (Are all such maximal sequences completely multiplicative?)
The longest discrepancy-2 sequence with Dirichlet inverse bounded by [math]\displaystyle{ 2 }[/math] has length [math]\displaystyle{ 389 }[/math]. Here is an example:
+1, -1, +1, +1, -1, -1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, -1, +1, +1, -1, +1, +1, -1, -1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, -1, +1, +1, -1, +1, +1, -1, -1, +1, -1, +1, +1, -1, +1, +1, -1, -1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, -1, +1, +1, -1, +1, +1, -1, -1, -1, -1, +1, +1, -1, +1, +1, -1, -1, +1, +1, -1, +1, -1, +1, +1, -1, -1, +1, -1, +1, +1, -1, +1, -1, -1, -1, +1, -1, -1, +1, -1, +1, +1, +1, -1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, -1, +1, -1, -1, +1, +1, +1, -1, +1, -1, +1, +1, -1, +1, -1, -1, -1, +1, -1, -1, +1, -1, +1, +1, +1, -1, -1, -1, +1, +1, -1, +1, +1, +1, -1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, -1, -1, +1, -1, +1, +1, -1, -1, -1, +1, +1, +1, -1, +1, +1, -1, -1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, +1, -1, -1, -1, +1, +1, -1, -1, +1, +1, +1, +1, -1, +1, -1, -1, -1, +1, -1, +1, +1, -1, +1, +1, -1, -1, +1, -1, -1, -1, +1, +1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, +1, -1, -1, -1, +1, +1, -1, +1, +1, -1, -1, +1, -1, -1, +1, +1, +1, +1, -1, -1, +1, -1, +1, -1, -1, +1, +1, -1, +1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, +1, -1, +1, -1, +1, -1, -1, -1, +1, -1, +1, +1, -1, +1, +1, -1, -1, +1, -1, +1, -1, -1, +1, +1, +1, -1, +1, -1, -1, +1, -1, -1, +1, +1, -1, +1, -1, +1, -1, -1, +1, +1, -1, -1, +1, -1, +1, +1, -1, +1, +1, -1, -1, +1, -1, -1, +1, -1, +1, -1, +1, +1, +1, -1, -1, +1, -1, +1, +1, -1, -1, +1, -1, -1, +1, -1, +1, +1, +1, -1, -1, +1, -1, +1, -1, +1, +1, -1, +1, -1, -1, +1, +1, -1, -1, +1, -1, +1, +1, -1, +1, -1, -1, -1, +1, +1, -1, +1, -1, -1, +1, +1
The longest discrepancy-2 sequence with Dirichlet inverse bounded by [math]\displaystyle{ 3 }[/math] has length at least [math]\displaystyle{ 489 }[/math].
