On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers
In this paper, the limit points of the sequence of arithmetic means <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn...
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2024-11-01
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| Series: | Mathematics |
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| author | Artūras Dubickas |
| author_facet | Artūras Dubickas |
| author_sort | Artūras Dubickas |
| collection | DOAJ |
| description | In this paper, the limit points of the sequence of arithmetic means <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><msubsup><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msup><mrow><mo>{</mo><msub><mi>H</mi><mi>m</mi></msub><mo>}</mo></mrow><mi>σ</mi></msup></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula> are studied, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math></inline-formula> is the <i>m</i>th harmonic number with fractional part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>H</mi><mi>m</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> is a fixed positive constant. In particular, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, it is shown that the largest limit point of the above sequence is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mo>(</mo><mi>e</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>0.581976</mn><mo>…</mo></mrow></semantics></math></inline-formula>, its smallest limit point is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><mo form="prefix">log</mo><mo>(</mo><mi>e</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>0.458675</mn><mo>…</mo></mrow></semantics></math></inline-formula>, and all limit points form a closed interval between these two constants. A similar result holds for the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><msubsup><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>f</mi><mrow><mo>(</mo><mrow><mo>{</mo><msub><mi>H</mi><mi>m</mi></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mi>σ</mi></msup></mrow></semantics></math></inline-formula> is replaced by an arbitrary absolutely continuous function <i>f</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-7f8f39edee6e4177b2c4bf2b06c39042 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-7f8f39edee6e4177b2c4bf2b06c390422024-12-13T16:27:34ZengMDPI AGMathematics2227-73902024-11-011223373110.3390/math12233731On the Range of Arithmetic Means of the Fractional Parts of Harmonic NumbersArtūras Dubickas0Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaIn this paper, the limit points of the sequence of arithmetic means <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><msubsup><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msup><mrow><mo>{</mo><msub><mi>H</mi><mi>m</mi></msub><mo>}</mo></mrow><mi>σ</mi></msup></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula> are studied, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math></inline-formula> is the <i>m</i>th harmonic number with fractional part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>H</mi><mi>m</mi></msub><mo>}</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> is a fixed positive constant. In particular, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, it is shown that the largest limit point of the above sequence is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mo>(</mo><mi>e</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>0.581976</mn><mo>…</mo></mrow></semantics></math></inline-formula>, its smallest limit point is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><mo form="prefix">log</mo><mo>(</mo><mi>e</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>0.458675</mn><mo>…</mo></mrow></semantics></math></inline-formula>, and all limit points form a closed interval between these two constants. A similar result holds for the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><msubsup><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mi>f</mi><mrow><mo>(</mo><mrow><mo>{</mo><msub><mi>H</mi><mi>m</mi></msub><mo>}</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>x</mi><mi>σ</mi></msup></mrow></semantics></math></inline-formula> is replaced by an arbitrary absolutely continuous function <i>f</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/12/23/3731harmonic numberfractional partarithmetic meanEuler’s constant |
| spellingShingle | Artūras Dubickas On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers Mathematics harmonic number fractional part arithmetic mean Euler’s constant |
| title | On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers |
| title_full | On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers |
| title_fullStr | On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers |
| title_full_unstemmed | On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers |
| title_short | On the Range of Arithmetic Means of the Fractional Parts of Harmonic Numbers |
| title_sort | on the range of arithmetic means of the fractional parts of harmonic numbers |
| topic | harmonic number fractional part arithmetic mean Euler’s constant |
| url | https://www.mdpi.com/2227-7390/12/23/3731 |
| work_keys_str_mv | AT arturasdubickas ontherangeofarithmeticmeansofthefractionalpartsofharmonicnumbers |