I am confused with what seems to be a standard notation in analytic number theory and I'd appreciate any clarification. I am interested in the zero density estimates, for example link.springer.com/article/10.1007/BF01403187 . In this paper and in many other sources I have seen, $N\_{\chi}(\alpha, T)$ is defined to be the number of zeros of $L(s, \chi)$ in the rectangle $\alpha \leq \sigma \leq 1$, $|t| \leq T$ where $\sigma$ is the real part of $s$.


What I was confused with is, is this the number of zeros counting with multiplicity (as in if $\rho$ is a zero of order $2$, then is $\rho$ counted twice) or not? All except one reference I have seen so far has no mention of multiplicity, so I thought it was counting without multiplicity. but it's not clear to me and I was hoping to get a clarification from an expert who knows this material.


This made me also wonder about in the notation
$$
\sum\_{\rho}
$$
where the sum is over the non-trivial zeroes of the L function, is this also sum taking into account the multiplicity (so $\sum\_{\rho} f(\rho)$ is actually
$$
\sum\_{\rho, distinct} (multiplicity \ of \ \rho) \ f(\rho) ?
$$
)
or not? For this one what I had in mind in particular was the explicit formula for $\sum\_{n \leq X} \chi(n) \Lambda(n)$. I have thought that it meant a sum without taking into account the multiplicity, but maybe this was incorrect.


It's not clear to me looking at the sources. Thanks in advance for the clarification. very appreciated.



Question about the notation $N_{\chi}(\alpha, T)$, the number of zeroes of the $L(s, \chi)$ in a rectangle
Zeros are always counted with multiplicity, both in $N\_\chi(\alpha,T)$ and in sums over zeros. This becomes clear when you look at how this quantity is estimated. Note also that the multiplicity of each zero $s$ of $L(s,\chi)$ is small, namely $O(\log q(2+|s|))$ by [Jensen's formula](https://en.wikipedia.org/wiki/Jensen%27s_formula).


For example, in Gallagher's paper the crucial step to look at is on p.336: "Since there are $\ll r\mathcal{L}$ zeros in the disc $|s-w|\leq r$ etc." This estimate comes from [Jensen's formula](https://en.wikipedia.org/wiki/Jensen%27s_formula), which counts the zeros with multiplicity. So in the next display, when $N\_\chi(\alpha,T)$ is bounded from above, zeros are taken into account with their multiplicities.


