Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.

Lets consider the sequence of induced characters $\chi^{P_N} $ obtained from $\chi_3$ having modulo $3.P_1...P_N$ where $P_N$ is the $N$-th prime number.

So at each step to define the new Character we replace by zero the value of multiple of $P_N$, we have :

$\chi_3 = (0,1,-1,0,1,-1,0,1,-1,...)$

$\chi^{P_1} =(0,1,0,0,0,-1,0,1,0,0,...)$ (all multiple of 2 have been put ot zero)

$\chi^{P_3} =(0,1,0,0,0,0,0,1,0,0,...)$ (all multiple of 5 have been put to zero)

So we obtain a sequence of induced characters with increasing modulo. My question is on the behavior of the maximum of the partial sum of the $\chi^{P_N}$ as $N$ increases :

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

$$Max_x(S(\chi^{P_N},x)$$

How this maximum will evoluate ?

My feeling is that this maximum will go up and down when $N$ goes to infinity and so that for a fixed constant $M$ (big enough) we can find an infinity of induced characters with this maximum lower than $M$. But is it true ? How to prove this is right or wrong ? Any reference on similar problem or idea ? Sieve theory can solve this kind of problem ?

(I already made a post on the subject in a more general way here : On partial sum of non-primitive Dirichlet characters)

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.

Lets consider the sequence of induced characters $\chi^{P_N} $ obtained from $\chi_3$ having modulo $3.P_1...P_N$ where $P_N$ is the $N$-th prime number.

So at each step to define the new Character we replace by zero the value of multiple of $P_N$, we have :

$\chi_3 = (0,1,-1,0,1,-1,0,1,-1,...)$

$\chi^{P_1} =(0,1,0,0,0,-1,0,1,0,0,...)$ (all multiple of 2 have been put ot zero)

$\chi^{P_3} =(0,1,0,0,0,0,0,1,0,0,...)$ (all multiple of 5 have been put to zero)

So we obtain a sequence of induced characters with increasing modulo. My question is on the behavior of the maximum of the partial sum of the $\chi^{P_N}$ as $N$ increases :

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

$$Max_x(S(\chi^{P_N},x)$$

How this maximum will evoluate ?

My feeling is that this maximum will go up and down when $N$ goes to infinity and so that for a fixed constant $M$ (big enough) we can find an infinity of induced characters with this maximum lower than $M$. But is it true ? How to prove this is right or wrong ? Any reference on similar problem or idea ? Sieve theory can solve this kind of problem ?

(I already made a post on the subject in a more general way here : On partial sum of non-primitive Dirichlet characters)

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.

Lets consider the suite of induced characters $\chi^{P_N} $ obtained from $\chi_3$ having modulo $3.P_1...P_N$ where $P_N$ is the $N$-th prime number.

So at each step to define the new Character we replace by zero the value of multiple of $P_N$, we have :

$\chi_3 = (0,1,-1,0,1,-1,0,1,-1,...)$

$\chi^{P_1} =(0,1,0,0,0,-1,0,1,0,0,...)$ (all multiple of 2 have been put ot zero)

$\chi^{P_3} =(0,1,0,0,0,0,0,1,0,0,...)$ (all multiple of 5 have been put to zero)

So we obtain a suite of induced characters with increasing modulo. My question is on the behavior of the maximum of the partial sum of the $\chi^{P_N}$ as $N$ increases :

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

$$Max_x(S(\chi^{P_N},x)$$

How this maximum will evoluate ?

My feeling is that this maximum will go up and down when $N$ goes to infinity and so that for a fixed constant $M$ (big enough) we can find an infinity of induced characters with this maximum lower than $M$. But is it true ? How to prove this is right or wrong ? Any reference on similar problem or idea ? Sieve theory can solve this kind of problem ?

(I already made a post on the subject in a more general way here : On partial sum of non-primitive Dirichlet characters)

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.

Lets consider the sequence of induced characters $\chi^{P_N} $ obtained from $\chi_3$ having modulo $3.P_1...P_N$ where $P_N$ is the $N$-th prime number.

So at each step to define the new Character we replace by zero the value of multiple of $P_N$, we have :

$\chi_3 = (0,1,-1,0,1,-1,0,1,-1,...)$

$\chi^{P_1} =(0,1,0,0,0,-1,0,1,0,0,...)$ (all multiple of 2 have been put ot zero)

$\chi^{P_3} =(0,1,0,0,0,0,0,1,0,0,...)$ (all multiple of 5 have been put to zero)

So we obtain a sequence of induced characters with increasing modulo. My question is on the behavior of the maximum of the partial sum of the $\chi^{P_N}$ as $N$ increases :

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

$$Max_x(S(\chi^{P_N},x)$$

How this maximum will evoluate ?

My feeling is that this maximum will go up and down when $N$ goes to infinity and so that for a fixed constant $M$ (big enough) we can find an infinity of induced characters with this maximum lower than $M$. But is it true ? How to prove this is right or wrong ? Any reference on similar problem or idea ? Sieve theory can solve this kind of problem ?

(I already made a post on the subject in a more general way here : On partial sum of non-primitive Dirichlet characters)

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.

Lets consider the suite of induced characters $\chi^{P_N} $ obtained from $\chi_3$ having modulo $3.P_1...P_N$ where $P_N$ is the N-th prime number.

So at each step to define the new Character we replace by zero the value of multiple of $P_N$, we have :

$\chi_3 = (0,1,-1,0,1,-1,0,1,-1,...)$

$\chi^{P_1} =(0,1,0,0,0,-1,0,1,0,0,...)$ (all multiple of 2 have been put ot zero)

$\chi^{P_3} =(0,1,0,0,0,0,0,1,0,0,...)$ (all multiple of 5 have been put to zero)

So we obtain a suite of induced characters with increasing modulo. My question is on the behavior of the maximum of the partial sum of the $\chi^{P_N}$ as $N$ increases :

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

$$Max_x(S(\chi^{P_N},x)$$

How this maximum will evoluate ?

My feeling is that this maximum will go up and down when $N$ goes to infinity and so that for a fixed constant $M$ (big enough) we can find an infinity of induced characters with this maximum lower than $M$. But is it true ? How to prove this is right or wrong ? Any reference on similar problem or idea ? Sieve theory can solve this kind of problem ?

(I already made a post on the subject in a more general way here : On partial sum of non-primitive Dirichlet characters)

Let's consider the Dirichlet Character $\chi_3(n)$ modulo 3 given by $\chi_3(1)=1$, $\chi_3(2)=-1$ and $\chi_3(3)=\chi_3(0)=0$.

Lets consider the suite of induced characters $\chi^{P_N} $ obtained from $\chi_3$ having modulo $3.P_1...P_N$ where $P_N$ is the $N$-th prime number.

So at each step to define the new Character we replace by zero the value of multiple of $P_N$, we have :

$\chi_3 = (0,1,-1,0,1,-1,0,1,-1,...)$

$\chi^{P_1} =(0,1,0,0,0,-1,0,1,0,0,...)$ (all multiple of 2 have been put ot zero)

$\chi^{P_3} =(0,1,0,0,0,0,0,1,0,0,...)$ (all multiple of 5 have been put to zero)

So we obtain a suite of induced characters with increasing modulo. My question is on the behavior of the maximum of the partial sum of the $\chi^{P_N}$ as $N$ increases :

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

$$Max_x(S(\chi^{P_N},x)$$

How this maximum will evoluate ?

My feeling is that this maximum will go up and down when $N$ goes to infinity and so that for a fixed constant $M$ (big enough) we can find an infinity of induced characters with this maximum lower than $M$. But is it true ? How to prove this is right or wrong ? Any reference on similar problem or idea ? Sieve theory can solve this kind of problem ?

(I already made a post on the subject in a more general way here : On partial sum of non-primitive Dirichlet characters)