Let's define $N(\alpha,T,\chi)=\sharp\lbrace \rho=\sigma+i\gamma: L(\rho,\chi)=0, \alpha\leq \sigma<1, |\gamma|\leq T\rbrace$ , where $\chi$ is a primitive Dirichlet character. We know, from Gallagher's paper, that

$$\sum_{q\leq T}\sum_{\chi\pmod{q}}N(\alpha,T,\chi)\ll T^{c(1-\alpha)}$$

for every $0\leq \alpha<1$ with $c>0$ an absolute costant.

But what about an explicit value of such costant $c$? I need some good and explicit estimate of the sum above. Can anyone help me?

  • $\begingroup$ According to Th. $10.4$ of the book by Iwaniec-Kowalski, one can take $c = \min(\frac{3}{2-\alpha},\frac{3}{3 \alpha - 1})$ (up to a logarithmic loss). Further bounds are given in the exercises of that section. However I do not know if someone has carried out a completely explicit such estimate. $\endgroup$ – js21 May 17 '17 at 15:30
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    $\begingroup$ Ramaré has some completely explicit zero-density estimates, e.g. in math.univ-lille1.fr/~ramare/Maths/DensityEstimate-29.pdf $\endgroup$ – Denis Chaperon de Lauzières May 17 '17 at 16:13
  • $\begingroup$ Habiba Kadiri cs.uleth.ca/~kadiri/research.html has several explicit estimates on zeroes, though not exactly of the form you are looking for. Still might be worth asking her directly though. $\endgroup$ – Terry Tao May 17 '17 at 17:10
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    $\begingroup$ For the sake of completeness, I would note that in the paper ''zeros of $L-$functions'' , Montgomery showed two estimates that one could rewrite easily in the form $\sum_{q\leq Q}\sum_{\chi\pmod{q}} N (\alpha,T,\chi)\ll (Q^{2}T)^{\frac{5}{2}(1-\alpha)}\log^{13}(QT), \forall \alpha$ with $0<\alpha<1$. $\endgroup$ – The Number Theorist May 18 '17 at 13:31

Matti Jutila (On Linnik's Constant, Math. Scand. 41 (1977), 45-62) proved that if $Q\geq 2$, $T\geq 1$, and $4/5\leq\alpha\leq 1$, and $\epsilon>0$, then

$\sum_{q\leq Q}\sideset{}{'}\sum_{\chi\bmod{q}}N(\alpha,\chi,T)\ll_{\epsilon}(Q^2 T)^{(2+\epsilon)(1-\alpha)},$

where the decoration ' indicates a sum over primitive characters. (The restriction to primitive characters is necessary.) The density hypothesis (which follows from the generalized Riemann hypothesis for Dirichlet $L$-functions) predicts that the above estimate holds for $1/2\leq\alpha\leq 1$, and so we have the density hypothesis in a limited range of $\alpha$. Montgomery's bound

$\sum_{q\leq Q}\sideset{}{'}\sum_{\chi\bmod q}N(\alpha,T,\chi)\ll(Q^2 T)^{\frac{3(1-\alpha)}{2-\alpha}}(\log QT)^{13},\qquad 1/2\leq\alpha\leq 4/5$

should suffice for most applications when $\alpha$ is far from 1.

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  • $\begingroup$ Is the assumption $Q \geq 2$ necessary? It doesn't seem to be present on the paper of Jutila? $\endgroup$ – Johnny T. Apr 19 '18 at 7:48
  • $\begingroup$ It is convenient to have $Q\geq 2$ in the event that you want to take $T=1$, in which case $\log(Q^2 T)>0$. But this is not absolutely necessary. For $Q=1$, this is simply a zero density estimate for the Riemann zeta function, which enjoys zero density estimates much stronger than these. $\endgroup$ – 2734364041 May 4 '18 at 19:47

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