Questions on explicit formulas related to $\frac{L_{k,j}(s)}{\zeta(s)}$

This question is related to explicit formulas for $$f_{k,j}(x)$$ defined in (1) below where $$\chi_{k,j}(n)$$ is a non-principal Dirichlet character.

(1) $$\quad f_{k,j}(x)=\sum\limits_{n=1}^x a_{k,j}(n)\,,\quad a_{k,j}(n)=\sum\limits_{d|n}\chi_{k,j}(d)\,\mu\left(\frac{n}{d}\right)$$

The Dirichlet transform of $$a_{k,j}(n)$$ defined in (1) above is defined in (2) below which I believe is valid for $$s\ge 1$$ (or $$s>\frac{1}{2}$$ assuming the generalized Riemann hypothesis).

(2) $$\quad F(s)=\sum\limits_{n=1}^\infty\frac{a_{k,j}(n)}{n^s}=\frac{L_{k,j}(s)}{\zeta(s)}\,,\quad\Re(s)\ge 1$$

The explicit formula for $$f(x)$$ defined in (1) above is defined in (2) below which I believe is valid for $$x>k$$ when $$\chi_{k,j}(n)$$ is a non-principal Dirichlet character. In some cases the constant term evaluates to zero, and in some cases the contribution of the trivial zeta zeros evaluates to zero.

(3) $$\quad \hat{f}_{k,j}(x)=-2\,L_{k,j}(0)+\sum_\limits{\rho}\frac{x^{\rho}\,L_{k,j}(\rho)}{\rho\,\zeta'(\rho)}+\sum\limits_n\frac{x^{-2 n}\,L_{k,j}(-2 n)}{-2 n\,\zeta'(-2 n)}$$

The explicit formula defined in (3) above is illustrated for several non-principal Dirichlet characters $$\chi_{k,j}(n)$$ following the questions below.

Question (1): Assuming $$\chi_{k,j}(n)$$ is a non-principal Dirichlet character, is it true in general that the explicit formula defined in (3) above is valid for $$x>k$$?

Question (2): What function is represented by the evaluation of the explicit formula for $$f_{5,3}(x)$$ in the interval $$1 (see Figure (3) below)?

The following figures illustrate $$\hat{f}_{k,j}(x)$$ defined in (3) above in orange overlaid on $$f_{k,j}(x)$$ defined in (1) above in blue where formula (3) is evaluated over the first $$100$$ pairs of non-trivial zeta zeros and $$30$$ trivial zeta zeros (except for $$\hat{f}_{5,3}(x)$$ illustrated in Figure (3) below which has no contribution from either the constant term or the trivial zeta zeros).

Figure (1): Illustration of $$\hat{f}_{3,2}(x)$$ where $$\chi_{3,2}(n)=\{1,-1,0\}$$

Figure (2): Illustration of $$\hat{f}_{4,2}(x)$$ where $$\chi_{4,2}(n)=\{1,0,-1,0\}$$

Figure (3): Illustration of $$\hat{f}_{5,3}(x)$$ where $$\chi_{5,3}(n)=\{1,-1,-1,1,0\}$$

Figure (4): Illustration of $$\Re\left(\hat{f}_{5,2}(x)\right)$$ where $$\chi_{5,2}(n)=\{1,i,-i,-1,0\}$$

Figure (5): Illustration of $$\Im\left(\hat{f}_{5,2}(x)\right)$$ where $$\chi_{5,2}(n)=\{1,i,-i,-1,0\}$$

I don't see the point of asking for the result. Don't you care of the maths ?

Look at the proof of the explicit formula for $$\zeta'(s)/\zeta(s)$$

see how it gets slightly more complicated for $$1/\zeta(s)$$,

adapt it to $$\zeta(s+1)/\zeta(s),\zeta(s)/\zeta(2s)$$, $$L(s,\chi)/\zeta(s)$$,

show that it fails for $$\zeta(s-1)/\zeta(s)$$ and $$\zeta(2s)/\zeta(s)$$, due to the growth as $$s\to 1/2+i\infty$$ and $$\Re(s) \to- \infty$$.

You'll find that the answer is the same as in your previous questions : $$\sum_\rho Res(L(s,\chi)/\zeta(s)\frac{x^s}{s},\rho)$$ converges when grouping the terms correctly.

The generalized Riemann hypothesis probably gives some bounds uniform in $$\chi$$ for the rate of convergence, as well as for all the corresponding explicit formulas where $$\zeta(s)$$ is replaced by $$L(s,\psi)$$.

• First I wanted to confirm the validity of the explicit formulas derived from $L_\chi(s)/\zeta(s)$ which you have. Jan 25, 2020 at 16:35
• I'd also like to understand what determines the convergence ranges of explicit formulas. The convergence ranges associated with the explicit formulas derived from $1/\zeta(s)$, $\zeta'(s)/\zeta(s)$, and $L_\chi(s)/\zeta(s)$ are different, and the convergence ranges of the explicit formulas derived from $L_\chi(s)/\zeta(s)$ (when $\chi$ is a non-principal Dirichlet character) seems to be a function of the modulus. It's not obvious to me from any of the answers exactly how these convergence ranges are determined. Jan 25, 2020 at 16:35
• Also, $L_{5,3}(s)/\zeta(s)$ converges to something in the range $1<x<5$, and I suspect there are other Dirichlet characters (possibly infinite) which exhibit this same phenomenon. I'd like to know what function these explicit formulas converge to in the range $1<x<k$ where $k$ is the modulus. Jan 25, 2020 at 16:35