Published 2025-01-01
“…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.…”
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