Timeline for Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?
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| Oct 9, 2021 at 16:24 | comment | added | Lucia | @HAHelfgott: No the constant is just that (basically the Euler product up to $(\log t)^2$). Take a look at the paper of Lamzouri, Li and Soundararajan. arxiv.org/pdf/1309.3595.pdf On page 4 you'll find an explicit bound stated (proved for characters in the paper because of the class number connectioon). | |
| Oct 9, 2021 at 9:38 | comment | added | H A Helfgott | @Lucia What is the tightest bound known on $1/\zeta(1+it)$ assuming RH, then? Is the constant better than $2e^\gamma \cdot 6/\pi^2$? | |
| Oct 9, 2021 at 9:37 | comment | added | H A Helfgott | @Lucia Ah, sorry, I misunderstood your answer. Granville-Soundararajan assumes GRH, but what you are suggesting at first is a $t$-analogue of that, and so it won't need RH. | |
| Oct 9, 2021 at 3:31 | comment | added | Lucia | @HAHelfgott: Hi Harald, I don't understand your question. In my answer for the questions on $\zeta$, only RH is used. What exactly are you asking about? | |
| Oct 8, 2021 at 7:57 | comment | added | H A Helfgott | Lucia (and @VesselinDimitrov) - How far down can we hope to go assuming RH but not GRH? Thm 14.9 of Titchmarsh (due to Littlewood) gives us an upper bound of $2 e^\gamma \cdot 6/pi^2$ instead of $2 e^\gamma$, assuming RH but not GRH, unless I am very mistaken. | |
| Jun 24, 2016 at 16:27 | comment | added | Vesselin Dimitrov | ...though it is less clear to me if the $O(1)$ should still be an absolute constant, as in the GRH bound, or must have a dependence on the degree. If it is absolute, and if this upper bound persists for all the partial sums, then the sum of $\Lambda(n)/n$ over the ideals $I \subset O_K$ of norm $n = N(I) \ll \log{|D|}\log{\log{|D|}}$ would be $\gg \log{\log{|D|}}$ with absolute implied constants, which is just what I'd need to improve Dobrowolski's bound on Mahler measures from $(\log{\log{d}}/\log{d})^3$ to $\gg 1/\log{d}$. | |
| Jun 24, 2016 at 16:23 | comment | added | Vesselin Dimitrov | Dear Lucia: thanks a lot! Regarding the logarithmic derivative case I was again thinking primarily of the Dedekind zeta of a general number field $K$ with discriminant $D$. For $L := \zeta_K / \zeta$, while $(L'/L)(1)$ can be very negative in the case of growing degree and small discriminant, Ihara proves in ["On the Euler-Kronecker constant..."] that the same type of GRH upper bound holds, uniformly in the degree: $(L'/L)(1) \leq 2\log{\log{|D|}} + O(1)$. Here too the expected bound will presumably be $\log{\log{|D|}}+\log{\log{\log{|D|}}} + O(1)$... (continued), | |
| Jun 24, 2016 at 15:29 | vote | accept | Vesselin Dimitrov | ||
| Jun 24, 2016 at 15:04 | history | edited | Lucia | CC BY-SA 3.0 |
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| Jun 24, 2016 at 14:54 | history | edited | Lucia | CC BY-SA 3.0 |
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| Jun 24, 2016 at 14:33 | history | answered | Lucia | CC BY-SA 3.0 |