Find in Library

The order of appearance <inline-formula> <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>:</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mrow> <mo>></mo> <mn>0</mn> </mrow> </msub> <mo>→</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mrow> <mo>></mo> <mn>0</mn> </mrow> </msub> </mrow> </semantics> </math> </inline-formula> is an arithmetic function related to the Fibonacci sequence <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>n</mi> </msub> </semantics> </math> </inline-formula>. This function is defined as the smallest positive integer solution of the congruence <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>F</mi> <mi>k</mi> </msub> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt"></mspace> <mrow> <mo>(</mo> <mi>mod</mi> <mspace width="0.277778em"></mspace> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. In this paper, we shall provide lower and upper bounds for the functions <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>∑</mo> <mrow> <mi>p</mi> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <mi>z</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>∑</mo> <mrow> <msup> <mi>p</mi> <mi>r</mi> </msup> <mo>≤</mo> <mi>x</mi> </mrow> </msub> <mi>z</mi> <mrow> <mo>(</mo> <msup> <mi>p</mi> <mi>r</mi> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>.