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This is kind of related to my previous question that was poorly stated because of misreading my own notes that I have taken on the papers I am currently reading, so no surprise that it eventually amounted to something trivial in prime number theory. (I am also surprised that it got upvoted so much!) I apologize for my clumsiness!

The real question that I had noted down and was meant to be asked is the following:

Let $\chi$ be a primitive character $\pmod k$. Then by Pólya's inequality we have $\forall x\geq 1$: $$\left|\sum_{n\leq x} \chi(n)\right|< \sqrt{k}\log k,$$ that is, the sum remains bounded when $x\to\infty$. My question is: what is the behavior of $$\left|\sum_{p\leq x} \chi(p)\right|,$$ when $x\to\infty$ ?

Notice that this is a whole different animal since with Pólya's inequality we had the periodicity of the Dirichlet characters over the natural numbers, whereas no such simple property holds if we restrict the sum to only prime numbers.

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    $\begingroup$ +1, very interesting question. I imagine it would help to decompose it as a linear combination of $\chi(a)\pi_{a,k}(x)$'s (with $\pi_{a,k}$ counting the number of primes with reside $a$ modulo $k$); obviously this is a rough measure of how 'imbalanced' the residues of primes are. Since $\sum \chi(a)=0$ it might be even more suggestive if we sliced out the average number of primes within a specific residue class, i.e. $$\left|\sum_{a\mod k}\chi(a)\left(\pi_{a,k}(x)-\frac{\pi(x)}{\phi(k)}\right)\right|.$$ (Edit: NB: this problem is as trivial as the last if $\chi$ is principal.) $\endgroup$
    – anon
    Commented Sep 27, 2011 at 0:48
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    $\begingroup$ Showing this sum is $O(x^{1/2+\epsilon})$ for any $\epsilon>0$ is exactly equivalent to the generalized Riemann hypothesis for Dirichilet L-functions (and one can show it is not smaller than $O(x^{1/2})$ by using that these functions have at least a zero with real part $\ge 1/2$). $\endgroup$
    – Rodrigo
    Commented Dec 25, 2017 at 2:53

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