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Let $d$ be a positive integer. Let $f(d,a)$ be the number of values of $x$ in $[1,d]$ such that $$x^{a}\equiv 1\pmod{d}.$$ I wanted to know if for some $0<\epsilon<1$, we can prove the following bound on the sum $$\sum_{i\mid\lambda(d)}\frac{f(d,i)}{i}\ll \lambda(d)^{\epsilon}$$ where $\lambda(d)$ is Carmichael's lambda function for almost all $d$. I found a resemblance of my question with Theorem 6 in this paper but couldn't translate those results to my setting.

Thanks in advance for any assistance.

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  • $\begingroup$ If $d=p_1\ldots p_k$ is a product of $k$ distinct primes and $i=2$ we get $f(d,2)=2^k$ and $\lambda(d)$ is lcm of numbers $p_i-1$, can not it be smaller than $2^k$? $\endgroup$ Nov 21, 2022 at 20:28
  • $\begingroup$ @FedorPetrov Sorry, I don't undertand your comment. Could you be more precise. $\endgroup$ Nov 21, 2022 at 20:32
  • $\begingroup$ What exactly is not clear? $\endgroup$ Nov 21, 2022 at 20:37
  • $\begingroup$ @FedorPetrov you are right. It has been proven that $\lambda(n)$ can take values as low as $(\log n)^{(\log\log\log n)}$ which would justify what you said. However, it is known that $\lambda(n)$ is quite large in a sequence of density 1 so I would edit my question. $\endgroup$ Nov 21, 2022 at 21:20

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