Timeline for Many representations as a sum of three squares
Current License: CC BY-SA 4.0
16 events
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| Feb 16, 2022 at 17:25 | comment | added | GH from MO | @MichaelBeeson Check out the paper that Lucia quoted. Theorem 5b says that for large $x$ there are at least $x^{1/10}$ square-free integers $d\leq x$ such that $L\bigl(1,\bigl(\frac{\cdot}{d}\bigr)\bigr)\geq e^\gamma(\log_2 x+\log_3 x-\log_4 x-10)$, where $\log_k x$ is the $k$ times iterated logarithm. | |
| Feb 16, 2022 at 5:44 | comment | added | Michael Beeson | Chowla and Walfisz both deal with large negative discriminants, showing there are plenty of small and large values respectively. What I want to know is that there are infinitely many discriminants (say d congruent to 1 mod 4) with $L(1,\chi) > c \sqrt d \log \log d$ for some explicit $c$. And anything else one can prove about such d. | |
| Sep 30, 2019 at 13:32 | comment | added | GH from MO | @PeterHumphries: Thanks, I fixed this! | |
| Sep 30, 2019 at 13:32 | history | edited | GH from MO | CC BY-SA 4.0 |
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| Sep 30, 2019 at 7:16 | comment | added | Peter Humphries | @GHfromMO, should the expression for $r_3(n)$ involve a sum over $d^2 \mid n$, not $d \mid n$? | |
| Aug 4, 2019 at 23:01 | history | edited | GH from MO | CC BY-SA 4.0 |
changed D to |D| in the bounds
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| Sep 9, 2015 at 20:51 | comment | added | Adam Sheffer | Thank you very much. Hopefully that's the end of these questions. | |
| Sep 9, 2015 at 20:40 | comment | added | GH from MO | @AdamSheffer: In fact Walfisz's paper concentrates on the negative fundamental discriminants. These are precisely the discriminants $D$ that arise from the square-free $n$'s with $n\equiv 1,2,3,5,6\pmod{8}$, see the precise formula for $D$ in my post. | |
| Sep 9, 2015 at 20:07 | comment | added | Adam Sheffer | There's something else that bothers me now (and I also wrote below the other answer). What if all of the $n$'s for which $L(1,\chi_d)$ is large are congruent to $0,4,7 \mod 8$? Then we still don't have the required bound for $r_3$. | |
| Sep 9, 2015 at 0:50 | comment | added | Adam Sheffer | Now it all makes sense :) | |
| Sep 8, 2015 at 23:46 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Sep 8, 2015 at 22:15 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Sep 8, 2015 at 22:05 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Sep 8, 2015 at 21:59 | comment | added | GH from MO | @AdamSheffer: It seems I got confused with the references. It was Walfisz (1942) who proved that Littlewood's upper bound is sharp (apart from the constant), while Chowla (1947) proved that Littlewood's lower bound is sharp (apart from the constant). I will update my post with more precise references. (BTW I don't know a direct connection between the existence of small $L$-values and the existence of large $L$-values.) | |
| Sep 8, 2015 at 21:35 | comment | added | Adam Sheffer | Thanks a lot! From looking at Chowla's paper (repository.ias.ac.in/8839/1/8839.pdf) it seems to me that he derived the opposite direction. That is, that there are many $n$'s with small $L$ values. Does this somehow imply a similar result for large value? | |
| Sep 7, 2015 at 22:46 | history | answered | GH from MO | CC BY-SA 3.0 |