I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?

Let's consider a real primitive Dirichlet character (not principal) $\chi$ of modulo q, and its L-function:

$$L(\chi,s)=\prod_{i=1}^{\infty} (1-\chi(p_i)p_i^{-s})^{-1}$$

For small $\epsilon >0$ we have $|L(\chi,\epsilon)| < M$.

We consider following sequence of L-functions ($p_i$ is the $i^{th}$ prime number):

$$L_N(\chi,s)= L(\chi,s) \prod_{i=1 ; P_i\nmid q}^{N} (1-\chi(p_i)p_i^{-s}) \prod_{j \in (b_1,...,b_k)} (1-\chi(p_j)p_j^{-s})$$

The $(b_1, ...b_k)$ are higher than $N$ and are chosen so that $\prod_{j \in (b_1,...,b_k)} (1-\chi(p_j)p_j^{-\epsilon})$ is sufficiently small (taking enough $p_{b_i}$ such that $\chi(p_{b_i})=1$) to have:

$$|L_N(\chi,\epsilon)|<M$$

Then we have a sequence of L-functions where we have "removed" the first $N$ primes plus some other primes (higher than $p_N$) so that the $L_N(\chi,s)$ functions have same zeros in the critical strip as $L(\chi,s)$ with their sum bounided by the same constant $M$.

Now each partial sum of the L-functions defined are bounded but can we find a general bound for all these partial sums? In other words can we find a constant $K$ such that :

Calling $a_{N,n}$ the coefficient of the Dirichlet $L_N(\chi,s)$ function (these coefficient are in fact coefficient of an induced charcter coming from $\chi$), so that:

$$L_N(\chi,s)= \sum_{n=1}^{\infty} \frac{a_{N,n}}{n^s}$$

We have for all $N$:

$$|\sum_{n<x} \frac{a_{N,n}}{n^{\epsilon}}| < K x^{\epsilon}$$

So is this type of bound finally as hard as GRH itself? Or is this type of bound not true? (note that choice of the $b_i$ provides flexibility to find a good sequence)

Perron's formula can maybe answer the question but up to now the bound I found are not satisfactory. Any idea ?

These partial sums area sums on integer free of small primes but I did not find any good reference. Also no reference on such sequence of L-function having same zeros and their properties. I made previous posts linked to this question:

Evolution of partial sum of a sequence of induced Dirichlet characters and