On partial sum of non-primitive Dirichlet characters

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 ?

• What does "Nieme prime" mean? All I've got so far is that Nieme is the name of a river. – Qiaochu Yuan Feb 26 '16 at 19:57
• @Qiaochu Yuan, what language did you pick for your reading exam? That is French for N-th: the N-th prime – KConrad Feb 26 '16 at 21:19
• The following version of the Pólya-Vinogradov bound holds for any non principal character $\chi \pmod{q}$: $\left| \sum_{M < n \leq M+N} \chi(n) \right| \leq 6\sqrt{q} \log q$ (cf. Iwaniec & Kowalski's Analytic Number Theory, p. 324). – José Hdz. Stgo. Feb 26 '16 at 22:49
• @KConrad: it was French, but in my defense the exam was very easy and I was allowed a French dictionary. Also, "nieme" is apparently Polish for "mute" as well. – Qiaochu Yuan Feb 27 '16 at 3:34
• Sorry, Nième is the N-th prime number... – Bertrand Feb 27 '16 at 9:35

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).

• Do you mean that $S(\chi,x)\geq \pi^{-1}\sqrt {q}$ is always true even for non primitive Dirichlet characters ? – Bertrand Feb 27 '16 at 9:40
• @Bertrand Definitely not, only for primitive characters. I meant that the bound was sharp for general $\chi$. I'll edit to make it clear. – Myshkin Feb 27 '16 at 9:48
• Thanks for your help, I found that for all characters we have $S(\chi,x) \ge \frac{\tau(\chi)}{\pi}$ (see Page 311) Multiplicative Number Theory I: Classical Theory Par Hugh L. Montgomery,Robert C. Vaughan – Bertrand Feb 27 '16 at 10:11
• In the case of the suite of induced characters mentionned in the question, using relation between Gauss sums of induced characters we have : $S(\chi^{P_N},x)\geq \pi^{-1} |\tau(\chi_1)|$, so the lower bound is a constant and we still do not know if the partial sums of an infinity of induced characters can be lower than a fixed constant. – Bertrand Feb 29 '16 at 10:15