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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 $1<x<5$ (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).


Illustration of formula (3) for k=3 and j=2

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


Illustration of formula (3) for k=4 and j=2

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


Illustration of formula (3) for k=5 and j=3

Illustration of formula (3) for k=5 and j=3

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


Illustration of real part of formula (3) for k=5 and j=2

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


Illustration of imaginary part of formula (3) for k=5 and j=2

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

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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 $L(s,\psi)$.

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  • $\begingroup$ First I wanted to confirm the validity of the explicit formulas derived from $L_\chi(s)/\zeta(s)$ which you have. $\endgroup$ Jan 25, 2020 at 16:35
  • $\begingroup$ 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. $\endgroup$ Jan 25, 2020 at 16:35
  • $\begingroup$ 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. $\endgroup$ Jan 25, 2020 at 16:35

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