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Questions tagged [multiplicative-function]

This tag is for questions relating to multiplicative functions which are arise most commonly in the field of number theory.

43 questions with no upvoted or accepted answers
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147 views

Minimal value of summatory function of completely multiplicative functions taking values -1 and 1

Here is a very nice paper http://www.ams.org/journals/tran/2010-362-12/S0002-9947-2010-05235-3/S0002-9947-2010-05235-3.pdf which led me to thinking about the problems below. Define the Liouville ...
386 views

Group-theoretic proof that an increasing multiplicative function is exponential

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing function such that $$f(2)=2$$ and $$f(mn)=f(m)f(n)$$ for all positive integers $m,n$ such that $$\gcd(m,n)=1$$ Prove that $f(n)=n$ ...
76 views

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime?

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime? Note, $f$ is multiplicative and $\sigma(n)>1, \;n>1$. Therefore $f(n)$ is prime only when $n=p^\alpha$, with $p$ prime, $\alpha\geq1$. ...
27 views

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Asymptotic estimate for the sum $\sum_{n\leq X}\mu(n)\tau(n)$?

Just trying to figure out what would be the asymptotic relation for the expression $\sum_{n\leq X}\mu(n)\tau(n)$, where $\tau$ corresponds to the number of divisors function (often named $\sigma_0$ ...
62 views

Interesting multiplicative Ramanujan-like q-expansions

We all know the full modular (cusp) form of weight 12 $$\Delta(z) = \sum_{n=1} \tau(n)q^n = q \prod_{n=1} (1-q^n)^{24}$$ that generates the multiplicative Ramunujan tau function $\tau(n)$. Today I ...
108 views

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Is there an analytic proof of change of bases in logarithms?

Usually change of bases in logarithms is just observance $$x=b^{\log_bx}\implies \log_kx={(\log_bx)}{(\log_kb)}\implies \log_kb=\frac{\log_kx}{\log_bx}.$$ Supposing apriori we do not know inverse of ...
26 views

$g(m)$ is a multiplicative fuction.

$g(m)=\sum_{n=1, (n,m)=1}^{n=m} e^{2\pi i n /m}$ I tried but didnt able to show it. Also how to show that $g(p)=-1$ for $p$ prime. I tried it just by splitting the series and collecting the terms ...
39 views

Does the Euler product stand for $a(n)=rad(n)$?

Does the Euler product stand for $a(n)=rad(n)$? Or more generally, for multiplicative functions which are not completely multiplicative? Where rad is the product of a number's distinct prime factors....