On partial sum of non-primitive Dirichlet characters

Question

Consider a Dirichlet character, $\chi(n)$, and the partial sum :

$$S(\chi,x)=\bigg |\sum_{n=1}^{x} \chi(n)\bigg|$$

There are many works to bound this sum when $\chi$ is a primitive character, but what can we say if $\chi$ is not primitive ?

More specifically if we fix a primitive Dirichlet character $\chi$ and then define from it a suite of non primitive character $\chi_N$ defined by:

$$\forall n, \chi_1(n)=\chi(n)$$

$$\forall n, \chi_N(n)=\chi_{N-1}(n).\chi^{P_N} (n)$$

Where $\chi^{P_N} (n)$ is the principal Dirichlet character associated to the N-th prime number (not considering 2).

(so $\chi^{P_N} (n)$ is the principal character simply defined by : $\chi^{P_N} (n) =0 $ if n is a multiple of the N-th prime number $P_N$ and 1 if not)

This suite of character is build from original character by "removing at each step a prime".

Question : how will evolute the max of $S(\chi^{P_N},x)$ for these charcaters ?

I would like to show that in this suite there are an infinity of characters with their $Max(S(\chi^{P_N},x))$ lower than a fix constant ? Is it realistic ?

Any reference on bounding partial sum of imprimitive characters ?

Answer 1

The original Pólya-Vinogradov inequality alredy works for non-primitive Dirichlet characters,

$$S(\chi,x)\leq c\sqrt {q} \log q$$

for some absolute constant $c$.

As J.H.S. mentions in the comments, apparently you can take $c=6$, see Iwaniec-Kowalski, theorem 12.5, p. 324, although they don't give a reference (I think that the Hildebrand improvement they mention is a stronger one, for primitive characters only).

Also, conditional on GRH, it is a result of Montgomery-Vaughan that

$$S(\chi,x)\ll \sqrt {q} \log\log q$$

Note that other than the GHR improvement, Pólya-Vinogradov is sharp, since $S(\chi,x)\geq \pi^{-1}\sqrt {q}$ for $\chi$ primitive. Also, $S(\chi,x)\gg \sqrt {q} \log\log q$ infinitely often for quadratic characters (Paley).