Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

We consider a wide class of summatory functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mfenced separators="" open="{" close="}"><mi>...

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Bibliographic Details
Main Author:	Leonid G. Fel
Format:	Article
Language:	English
Published:	MDPI AG 2025-01-01
Series:	Mathematics
Subjects:	multiplicative number theory summatory functions asymptotic analysis
Online Access:	https://www.mdpi.com/2227-7390/13/2/281
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Summary:	We consider a wide class of summatory functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mfenced separators="" open="{" close="}"><mi>f</mi><mo>;</mo><mi>N</mi><mo>,</mo><msup><mi>p</mi><mi>m</mi></msup></mfenced><mo>=</mo><msub><mo>∑</mo><mrow><mi>k</mi><mo>≤</mo><mi>N</mi></mrow></msub><mi>f</mi><mfenced separators="" open="(" close=")"><msup><mi>p</mi><mi>m</mi></msup><mi>k</mi></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub><mo>∪</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula> associated with the multiplicative arithmetic functions <i>f</i> of a scaled variable <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>, where <i>p</i> is a prime number. Assuming an asymptotic behavior of the summatory function, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mrow><mo>{</mo><mi>f</mi><mo>;</mo><mi>N</mi><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mover><mo>=</mo><mrow><mi>N</mi><mo>→</mo><mo>∞</mo></mrow></mover><msub><mi>G</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mfenced separators="" open="[" close="]"><mn>1</mn><mo>+</mo><mi mathvariant="script">O</mi><mfenced separators="" open="(" close=")"><msub><mi>G</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mfenced></mfenced></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>N</mi><msub><mi>a</mi><mn>1</mn></msub></msup><msup><mfenced separators="" open="(" close=")"><mi>log</mi><mi>N</mi></mfenced><msub><mi>b</mi><mn>1</mn></msub></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>N</mi><mrow><mo>−</mo><msub><mi>a</mi><mn>2</mn></msub></mrow></msup><msup><mfenced separators="" open="(" close=")"><mi>log</mi><mi>N</mi></mfenced><mrow><mo>−</mo><msub><mi>b</mi><mn>2</mn></msub></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mo>∞</mo><mo><</mo><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, we calculate the renormalization function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mfenced separators="" open="(" close=")"><mi>f</mi><mo>;</mo><mi>N</mi><mo>,</mo><msup><mi>p</mi><mi>m</mi></msup></mfenced></mrow></semantics></math></inline-formula>, defined as a ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mfenced separators="" open="{" close="}"><mi>f</mi><mo>;</mo><mi>N</mi><mo>,</mo><msup><mi>p</mi><mi>m</mi></msup></mfenced><mo>/</mo><mi>F</mi><mrow><mo>{</mo><mi>f</mi><mo>;</mo><mi>N</mi><mo>,</mo><mn>1</mn><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, and find its asymptotics <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mo>∞</mo></msub><mfenced separators="" open="(" close=")"><mi>f</mi><mo>;</mo><msup><mi>p</mi><mi>m</mi></msup></mfenced></mrow></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. We prove that a renormalization function is multiplicative, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mo>∞</mo></msub><mfenced separators="" open="(" close=")"><mi>f</mi><mo>;</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mi>p</mi><mi>i</mi><msub><mi>m</mi><mi>i</mi></msub></msubsup></mfenced><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>R</mi><mo>∞</mo></msub><mfenced separators="" open="(" close=")"><mi>f</mi><mo>;</mo><msubsup><mi>p</mi><mi>i</mi><msub><mi>m</mi><mi>i</mi></msub></msubsup></mfenced></mrow></semantics></math></inline-formula> with <i>n</i> distinct primes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>p</mi><mi>i</mi></msub></semantics></math></inline-formula>. We extend these results to the other summatory functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∑</mo><mrow><mi>k</mi><mo>≤</mo><mi>N</mi></mrow></msub><mi>f</mi><mrow><mo>(</mo><msup><mi>p</mi><mi>m</mi></msup><msup><mi>k</mi><mi>l</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>l</mi><mo>,</mo><mi>k</mi><mo>∈</mo><msub><mi mathvariant="double-struck">Z</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∑</mo><mrow><mi>k</mi><mo>≤</mo><mi>N</mi></mrow></msub><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>f</mi><mi>i</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><msup><mi>p</mi><msub><mi>m</mi><mi>i</mi></msub></msup></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>≠</mo><msub><mi>f</mi><mi>j</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub><mo>≠</mo><msub><mi>m</mi><mi>j</mi></msub></mrow></semantics></math></inline-formula>. We apply the derived formulas to a large number of basic summatory functions including the Euler <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϕ</mi><mo>(</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> and Dedekind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ψ</mi><mo>(</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> totient functions, divisor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>σ</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and prime divisor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi><mo>(</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula> functions, the Ramanujan sum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>q</mi></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and Ramanujan <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> Dirichlet series, and others.
ISSN:	2227-7390