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# BoundingTheCorrelationof the PhiMobius Function Twistedbyaand Dirichlet CharacterCharacters.

Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$

In other words

$$\phi_{\chi}(n)=\sum_{d|n}\mu\left(\frac{n}{d}\right)\chi\left(\frac{n}{d}\right)d=\left(\text{Id}*\mu\chi\right)(n).$$

My question is, how large can $\phi_\chi(n)$ be? More precisely, what is the smallest function $f$ such that for all $n$ and $\chi$ $$\frac{\phi_{\chi}(n)}{n}=\sum_{d|n}\frac{\mu(d)\chi(d)}{d}\ll f(n).$$

It is not hard to see that $\frac{\phi_{\chi}(n)}{n}\ll \log n$ for all $n$ so $f\ll \log n$. For Euler's $\phi$ function, the first term is the main contributor, as $\phi(n)\leq n$ we know that $\frac{\phi(n)}{n}\leq 1$ for all $n$. Since $\chi$ doesn't add to muchhas norm $1$, and the sums $\sum_{n\leq x}\mu(n)\chi(n)$ are small, we might conjecture that $\frac{\phi_\chi (n)}{n}\leq 1$.

Remarkably,

However this is not so. We can find a character such that $\mu$ and $\chi$ function have a lot of correlation, enough to make the sum of size $\sqrt{\log\log n}$. This is outlined by the following construction:

Let $n$ be the product of all primes $p=3+4k$ where $p\leq M$, and let $\chi$ be the unique quadratic character modulo $4$. Then choose whether or not to remove the prime $3$ from this product as to force the equivalence $n\equiv 1 \pmod{4}$. For each divisor $d$ of $n$, if $\omega(d)$ is even, then $d\equiv 1\pmod{4}$ so that $\chi(d)\mu(d)=1$, and if $\omega(d)$ is odd, then $d\equiv 3\pmod{4}$ so that $\chi(d)\mu(d)=1$ yet again. This means that $$\frac{\phi_\chi (n)}{n}=\sum_{d|n} \frac{1}{d}\gg \sqrt{\log M}\gg\sqrt{\log \log n}.$$ The second last $\gg$ follows from properties the fact that if $A$ is the set of those integers composed only of primes congruent to $3$ modulo $4$.4$, then$\sum_{n\leq M,\ n\in A} \frac{1}{n}=\sqrt{\log M}$, and the last$\gg$follows from the fact that$\log n =\theta(M;4,3)$. Any references to papers which might deal with this sort of sum is greatly appreciated, Thanks, 1 # Bounding the Phi Function Twisted by a Dirichlet Character Let$\chi$be a Dirichlet character, and define$\phi_\chi (n)$so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$ In other words $$\phi_{\chi}(n)=\sum_{d|n}\mu\left(\frac{n}{d}\right)\chi\left(\frac{n}{d}\right)d=\left(\text{Id}*\mu\chi\right)(n).$$ My question is, how large can$\phi_\chi(n)$be? More precisely, what is the smallest function$f$such that for all$n$and$\chi$$$\frac{\phi_{\chi}(n)}{n}=\sum_{d|n}\frac{\mu(d)\chi(d)}{d}\ll f(n).$$ It is not hard to see that$\frac{\phi_{\chi}(n)}{n}\ll \log n$for all$n$so$f\ll \log n$. For Euler's$\phi$function, the first term is the main contributor, as$\phi(n)\leq n$we know that$\frac{\phi(n)}{n}\leq 1$for all$n$. Since$\chi$doesn't add to much, we might conjecture$\frac{\phi_\chi (n)}{n}\leq 1$. Remarkably, this is not so. We can find a character such that$\mu$and$\chi$function have a lot of correlation, enough to make the sum of size$\sqrt{\log\log n}$. This is outlined by the following construction: Let$n$be the product of all primes$p=3+4k$where$p\leq M$, and let$\chi$be the unique quadratic character modulo$4$. Then choose whether or not to remove the prime$3$from this product as to force the equivalence$n\equiv 1 \pmod{4}$. For each divisor$d$of$n$, if$\omega(d)$is even, then$d\equiv 1\pmod{4}$so that$\chi(d)\mu(d)=1$, and if$\omega(d)$is odd, then$d\equiv 3\pmod{4}$so that$\chi(d)\mu(d)=1$yet again. This means that $$\frac{\phi_\chi (n)}{n}=\sum_{d|n} \frac{1}{d}\gg \sqrt{\log M}\gg\sqrt{\log \log n}.$$ The second last$\gg$follows from properties of those integers composed only of primes congruent to$3$modulo$4\$.

Any references to papers which might deal with this sort of sum is greatly appreciated,

Thanks,