Timeline for The mean value of $y \log{y}$ over the ordinates of the CM points
Current License: CC BY-SA 3.0
8 events
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| Jul 1, 2017 at 19:52 | history | edited | Peter Humphries | CC BY-SA 3.0 |
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| Jun 28, 2017 at 2:28 | history | edited | Peter Humphries | CC BY-SA 3.0 |
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| Jun 27, 2017 at 22:37 | comment | added | Vesselin Dimitrov | [Reg. the edit,to rectify my comment: While $L(1) = o(1)$ and $(L'/L)(1) \sim 1/(1-\beta)$ under the hypothesis of a Siegel zero with $(1-\beta) \log{|D|} \to 0$, I suppose it is $(1-\beta)(L''/L)(1)$ whose magnitude rel. $\log{|D|}$ may not 'a priori' be determined - unless e.g. one assumes the remaining non-trivial zeros to lie on the critical line. In contrast to the case $g(z) = y$, this means that the average value for $g(z) = y\log{y}$ may not be asymptotically computed as a function of $D$ and $\beta$ under the sole hypothesis of such a Siegel zero $\beta$.] | |
| Jun 27, 2017 at 21:42 | history | edited | Peter Humphries | CC BY-SA 3.0 |
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| Jun 27, 2017 at 21:13 | vote | accept | Vesselin Dimitrov | ||
| Jun 27, 2017 at 21:11 | comment | added | Vesselin Dimitrov | Many thanks for this detailed and instructive answer! Yes, according to Iwaniec and Friedlander (a recent reference below), not even the trivial bound $L'(1,\chi_D) \ll (\log{|D|})^2$ has been unconditionally improved to a $o(\cdot)$. This also explains my 'observation' that no asymptotic may be given under the extraordinary hypothesis of a Siegel zero, for then, unless one assumes something supplementary for the complex zeros, the derivative $L'(1,\chi_D)$ may a priori be anything in the range $1 \ll R \ll (\log{|D|})^2$ (cf. Iwaniec and Friedlander's Note on Dirichlet $L$-functions). | |
| Jun 27, 2017 at 19:23 | history | edited | Peter Humphries | CC BY-SA 3.0 |
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| Jun 27, 2017 at 18:52 | history | answered | Peter Humphries | CC BY-SA 3.0 |