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I'm reading Tom M. Apostol's introduction to analytic number theory. In chapter 2 he defines

derivative of arithmetical function

Does his definition apply to this modular polynomial in the question? I feel it does not. I'd think the derivative of the function in the question is just $f'(n) = 2n$. That's the formal derivative. I'm interested in knowing what $f'(n)$ is like in his theorem 5.30. What derivative is that?

Theorem 5.30

user851954
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    Since Apostol has called it a definition, it need not match what you call a formal derivative. – coffeemath May 14 '21 at 19:51
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    This is a different kind of derivative. You can’t apply the real derivative to arithmetical functions, because they are only defined on $\mathbb N^+,$ so the real definition of derivative as a limit cannot be applied to arithmetical functions. – Thomas Andrews May 14 '21 at 19:52
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  • If I apply his definition to the polynomial in the question, I get $f'(n) = f(n) \log(n)$, so $f'(5) = 16\log(5)$, which is not an integer. – user851954 May 14 '21 at 19:59
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    By the way, if $g(n)=(17n^2-9)\bmod 16,$ you’d have $g’(n)=34n,$ but $g(n)=f(n),$ so your definition of derivative would be not of a function, but of its written representation. – Thomas Andrews May 14 '21 at 20:45

1 Answers1

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The arithmetic functions are only defined on $\mathbb N^+,$ so you can’t use the calculus limit definition to get a standard derivative.

So in your case, you just apply the definition:

$$f’(n)=(n^2-9\bmod 16)\log n$$

The relation to calculus definitions is if $f$ is arithmetical, then there is a related function, $F,$ (called a Dirichlet series):

$$F(t)=\sum_{n=1}^\infty f(n)n^t$$

Usually defined for $t$ with a negative upper bound on the real part, but you can also think of $F$ as an abstract series, the way you can think of power series abstractly. (Dirichlet series are usually defined in terms of $s,$ where $t=-s,$ but I chose this version because it otherwise changes the sign of $F’.$)

Then:

$$F’(t)=\sum_{n=1}^\infty \log(n)\cdot f(n)n^t=\sum_{n=1}^\infty f’(n)n^t$$

where $F’$ is the usual calculus/complex analysis derivative, and $f’$ is Apostol’s definition for arithmetical functions.

The relation between $F$ and $f$ is crucial in studying arithmetical functions. Given $f,g$ arithmetical and corresponding Dirichlet series $F,G,$ the Dirichlet series for $f+g$ is $F+G,$ and the Dirichlet series of $h=f*g,$ where:

$$(f*g)(n)=\sum_{d\mid n}f(d)g(n/d)$$ has corresponding Dirichlet series, $H(t)=F(t)G(t).$

On thing you get from this correspondence is the usual identities:$$(\alpha f)’=\alpha\cdot (f’)\\(f+g)’=f’+g’\\(f*g)’=f’*g +f*g’.$$ where $\alpha$ is a constant.

Thomas Andrews
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    Re: final paragraph: if $Lf := Df/f,$ then $,L(fg) = Lf + Lg\iff D(fg) = fDg + gDf,,$ cf. logarithmic differentiation – Bill Dubuque May 14 '21 at 21:30
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    Not sure how that applies here. Is $fg$ the same as $fg?$ Is $Df/f$ actual function division, or $$-multiplication by the $$-inverse of $f?$ If $f(1)=0$ there is no $$-inverse of $f.$ The actual result follows directly from $\log(n)=\log(d)+\log(n/d)$ so this seems like overkill. @BillDubuque – Thomas Andrews May 14 '21 at 22:04