View source for Polymath15 test problem

We are initially focusing attention on the following 

:'''Test problem'''  For <math>t=y=0.4</math>, can one prove that <math>H_t(x+iy) \neq 0</math> for all <math>x \geq 0</math>?

If we can show this, it is likely that (with the additional use of the argument principle, and some further information on the behaviour of <math>H_t(x+iy)</math> at <math>y=0.4</math>) that one can show that <math>H_t(x+iy) \neq 0</math> for all <math>y \geq 0.4</math> as well.  This would give a new upper bound

:<math>\Lambda \leq 0.4 + \frac{1}{2}  (0.4)^2 = 0.48</math>

for the de Bruijn-Newman constant.

For very small values of <math>x</math> we expect to be able to establish this by direct calculation of <math>H_t(x+iy)</math>.  For medium or large values, the strategy is to use a suitable approximation

:<math>H_t(x+iy) \approx A + B</math>

for some relatively easily computable quantities <math>A = A_t(x+iy), B = B_t(x+iy)</math> (it may possibly be necessary to use a refined approximation <math>A+B-C</math> instead).  The quantity <math>B</math> contains a non-zero main term <math>B_0</math> which is expected to roughly dominate.  To show <math>H_t(x+iy)</math> is non-zero, it would suffice to show that

:<math> \frac{|H_t - A - B|}{|B_0|} < \frac{|A + B|}{|B_0|}.</math>

Thus one will seek upper bounds on the error <math>\frac{|H_t - A - B|}{|B_0|}</math> and lower bounds on <math>\frac{|A+B|}{|B_0|}</math> for various ranges of <math>x</math>.  Numerically it seems that the RHS stays above 0.4 as soon as <math>x</math> is moderately large, while the LHS stays below 0.1, which looks promising for the rigorous arguments.

== Choices of approximation ==

There are a number of slightly different approximations we have used in previous discussion.  The first approximation was <math>A+B</math>, where

:<math>A := \frac{1}{8} \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \sum_{n=1}^N \frac{\exp(\frac{t}{16} \log^2 \frac{s+4}{2\pi n^2})}{n^s}</math>
:<math>B := \frac{1}{8} \frac{s(s-1)}{2} \pi^{-(1-s)/2} \Gamma((1-s)/2) \sum_{n=1}^N \frac{\exp(\frac{t}{16} \log^2 \frac{5-s}{2\pi n^2})}{n^{1-s}}</math>
:<math>B_0 := \frac{1}{8} \frac{s(s-1)}{2} \pi^{-(1-s)/2} \Gamma((1-s)/2) \exp( \frac{t}{16} \log^2 \frac{5-s}{2\pi} )</math>
:<math>s := \frac{1-y+ix}{2}</math>
:<math>N := \lfloor \sqrt{\frac{\mathrm{Im} s}{2\pi}} \rfloor = \lfloor \sqrt{\frac{x}{4\pi}} \rfloor.</math>

There is also the refinement <math>A+B-C</math>, where
:<math> C:= \frac{1}{8} \exp(-\frac{t\pi^2}{64}) \frac{s(s-1)}{2} \pi^{-s/2} \Gamma(s/2) \frac{e^{-i\pi s} \Gamma(1-s)}{2\pi i} (2\pi i N)^{s-1} \Psi( \frac{s}{2\pi i N}-N )</math>
:<math> \Psi(\alpha) := 2\pi \frac{\cos \pi(\frac{1}{2} \alpha^2 - \alpha - \frac{\pi}{8})}{\cos(\pi \alpha)} \exp( \frac{i \pi}{2} \alpha^2 - \frac{5 \pi i}{8}).</math>

The first approximation was modified slightly to <math>A'+B'</math>, where

:<math>A' := \frac{2}{8} \pi^{-s/2} \sqrt{2\pi} \exp( (\frac{s+4}{2}-\frac{1}{2}) \log \frac{s+4}{2} - \frac{s+4}{2}) \sum_{n=1}^N \frac{\exp(\frac{t}{16} \log^2 \frac{s+4}{2\pi n^2})}{n^s}</math>
:<math>B' := \frac{2}{8} \pi^{-(1-s)/2} \sqrt{2\pi} \exp( (\frac{5-s}{2}-\frac{1}{2}) \log \frac{5-s}{2} - \frac{5-s}{2}) \sum_{n=1}^N \frac{\exp(\frac{t}{16} \log^2 \frac{5-s}{2\pi n^2})}{n^{1-s}}</math>
:<math>B'_0 := \frac{2}{8} \pi^{-(1-s)/2} \sqrt{2\pi} \exp( (\frac{5-s}{2}-\frac{1}{2}) \log \frac{5-s}{2} - \frac{5-s}{2}) \exp( \frac{t}{16} \log^2 \frac{5-s}{2\pi} ) </math>
:<math>s := \frac{1-y+ix}{2}</math>
:<math>N := \lfloor \sqrt{\frac{\mathrm{Im} s}{2\pi}} \rfloor = \lfloor \sqrt{\frac{x}{4\pi}} \rfloor.</math>

In [[Effective bounds on H_t - second approach]], a more refined approximation <math>A^{eff} + B^{eff}</math> was introduced:

:<math> A^{eff} := \frac{1}{8} \exp( \frac{t}{4} \alpha_1(\frac{1-y+ix}{2})^2 ) H_{0,1}(\frac{1-y+ix}{2}) \sum_{n=1}^N \frac{1}{n^{\frac{1-y+ix}{2} + \frac{t \alpha_1(\frac{1-y+ix}{2})}{2} - \frac{t}{4} \log n}}</math>
:<math> B^{eff} := \frac{1}{8} \exp( \frac{t}{4} \overline{\alpha_1(\frac{1+y+ix}{2})}^2 ) \overline{H_{0,1}(\frac{1+y+ix}{2})} \sum_{n=1}^N \frac{1}{n^{\frac{1+y-ix}{2} + \frac{t \overline{\alpha_1(\frac{1+y+ix}{2})}}{2} - \frac{t}{4} \log n}}</math>
:<math> B^{eff}_0 := \frac{1}{8} \exp( \frac{t}{4} \overline{\alpha_1(\frac{1+y+ix}{2})}^2 ) \overline{H_{0,1}(\frac{1+y+ix}{2})} </math>
:<math>H_{0,1}(s) := \frac{s (s-1)}{2} \pi^{-s/2} \sqrt{2\pi} \exp( (\frac{s}{2} - \frac{1}{2}) \log \frac{s}{2} - \frac{s}{2} )</math>
:<math> \alpha_1(s) := \frac{1}{2s} + \frac{1}{s-1} + \frac{1}{2} \log \frac{s}{2\pi} </math>
:<math> N := \lfloor \sqrt{ \frac{T'}{2\pi}} \rfloor</math>
:<math> T' := \frac{x}{2} + \frac{\pi t}{8}.</math>

There is a refinement <math>A^{eff}+B^{eff}-C^{eff}</math>, where
:<math>C^{eff} := \frac{1}{8} \exp( \frac{t\pi^2}{64}) \frac{s'(s'-1)}{2} (-1)^N ( \pi^{-s'/2} \Gamma(s'/2) a^{-\sigma} C_0(p) U + \pi^{-(1-s')/2} \Gamma((1-s')/2) a^{-(1-\sigma)} \overline{C_0(p)} \overline{U})</math>
:<math>s' := \frac{1-y}{2} + iT' = \frac{1-y+ix}{2} + \frac{\pi i t}{8} </math>
:<math>a := \sqrt{\frac{T'}{2\pi}}</math>
:<math>p := 1 - 2(a-N)</math>
:<math>\sigma := \mathrm{Re} s' = \frac{1-y}{2}</math>
:<math>U := \exp( -i (\frac{T'}{2} \log \frac{T'}{2\pi} - \frac{T'}{2} - \frac{\pi}{8} ))</math>
:<math>C_0(p) := \frac{ \exp( \pi i (p^2/2 + 3/8) )- i \sqrt{2} \cos(\pi p/2)}{2 \cos(\pi p)}.</math>

Finally, a simplified approximation is <math>A^{toy} + B^{toy}</math>, where

:<math> A^{toy} := B^{toy}_0 \exp(i ((\frac{x}{2} + \frac{\pi t}{8}) \log \frac{x}{4\pi} - \frac{x}{2} - \frac{\pi}{4} )) N^{-y} \sum_{n=1}^N \frac{1}{n^{\frac{1-y+ix}{2} + \frac{t}{4} \log \frac{N^2}{n} + \pi i t/8}}</math>
:<math> B^{toy} := B^{toy}_0 \sum_{n=1}^N \frac{1}{n^{\frac{1+y-ix}{2} + \frac{t}{4} \log \frac{N^2}{n} - \pi i t/8}}</math>
:<math> B^{toy}_0 := \frac{\sqrt{2}}{4} \pi^2 N^{\frac{7+y}{2}} \exp( i (-\frac{x}{4} \log \frac{x}{4\pi} + \frac{x}{4} + \frac{9-y}{8} \pi) + \frac{t}{16} (\log \frac{x}{4\pi} - \frac{\pi i}{2})^2 ) e^{-\pi x/8} </math>
:<math> N := \lfloor \sqrt{\frac{x}{4\pi}} \rfloor.</math>

Here is a table comparing the size of the various main terms:

{| border=1
|-
! style="text-align:left;"| <math>x</math>
! <math>B_0</math>
! <math>B'_0</math>
! <math>B^{eff}_0</math>
! <math>B^{toy}_0</math>
|-
| <math>10^3</math>
| <math>(3.4405 + 3.5443 i) \times 10^{-167}</math>
| <math>(3.4204 + 3.5383 i) \times 10^{-167}</math>
| <math>(3.4426 + 3.5411 i) \times 10^{-167}</math>
| <math>(2.3040 + 2.3606 i) \times 10^{-167}</math>
|-
| <math>10^4</math>
| <math>(-1.1843 - 7.7882 i) \times 10^{-1700}</math>
| <math>(-1.1180 - 7.7888 i) \times 10^{-1700}</math>
| <math>(-1.1185 - 7.7879 i) \times 10^{-1700}</math>
| <math>(-1.1155 - 7.5753 i) \times 10^{-1700}</math>
|-
| <math>10^5</math>
| <math>(-7.6133 + 2.5065 i) * 10^{-17047} </math>
| <math>(-7.6134 + 2.5060 i) * 10^{-17047} </math>
| <math>(-7.6134 + 2.5059 i) * 10^{-17047} </math>
| <math>(-7.5483 + 2.4848 i) * 10^{-17047} </math>
|-
| <math>10^6</math>
| <math>(-3.1615 - 7.7093 i) * 10^{-170537} </math>
| <math>(-3.1676 - 7.7063 i) * 10^{-170537} </math>
| <math>(-3.1646 - 7.7079 i) * 10^{-170537} </math>
| <math>(-3.1590 - 7.6898 i) * 10^{-170537} </math>
|-
| <math>10^7</math>
| <math>(2.1676 - 9.6330 i) * 10^{-1705458} </math>
| <math>(2.1711 - 9.6236 i) * 10^{-1705458} </math>
| <math>(2.1571 - 9.6329 i) * 10^{-1705458} </math>
| <math>(2.2566 - 9.6000 i) * 10^{-1705458} </math>
|}

Here some typical values of <math>B/B_0</math> (note that <math>B/B_0</math> and <math>B'/B'_0</math> are identical):

{| border=1
|-
! style="text-align:left;"| <math>x</math>
! <math>B/B_0</math>
! <math>B'/B'_0</math>
! <math>B^{eff}/B^{eff}_0</math>
! <math>B^{toy}/B^{toy}_0</math>
|-
| <math>10^3</math>
| <math>0.7722 + 0.6102 i</math>
| <math>0.7722 + 0.6102 i</math>
| <math>0.7733 + 0.6101 i</math>
| <math>0.7626 + 0.6192 i</math>
|-
| <math>10^4</math>
| <math>0.7434 - 0.0126 i</math>
| <math>0.7434 - 0.0126 i</math>
| <math>0.7434 - 0.0126 i</math>
| <math>0.7434 - 0.0124 i</math>
|-
| <math>10^5</math>
| <math>1.1218 - 0.3211 i</math>
| <math>1.1218 - 0.3211 i </math>
| <math>1.1218 - 0.3211 i </math>
| <math>1.1219 - 0.3213 i </math>
|-
| <math>10^6</math>
| <math>1.3956 - 0.5682 i</math>
| <math>1.3956 - 0.5682 i</math>
| <math>1.3955 - 0.5682 i</math>
| <math>1.3956 - 0.5683 i</math>
|-
| <math>10^7</math>
| <math>1.6400 + 0.0198 i</math>
| <math>1.6400 + 0.0198 i</math>
| <math>1.6401 + 0.0198 i</math>
| <math>1.6400 - 0.0198 i</math>
|}

Here some typical values of <math>A/B_0</math>, which seems to be about an order of magnitude smaller than <math>B/B_0</math> in many cases:

{| border=1
|-
! style="text-align:left;"| <math>x</math>
! <math>A/B_0</math>
! <math>A'/B'_0</math>
! <math>A^{eff}/B^{eff}_0</math>
! <math>A^{toy}/B^{toy}_0</math>
|-
| <math>10^3</math>
| <math>-0.3856 - 0.0997 i</math>
| <math>-0.3857 - 0.0953 i</math>
| <math>-0.3854 - 0.1002 i</math>
| <math>-0.4036 - 0.0968 i</math>
|-
| <math>10^4</math>
| <math>-0.2199 - 0.0034 i</math>
| <math>-0.2199 - 0.0036 i</math>
| <math>-0.2199 - 0.0033 i</math>
| <math>-0.2208 - 0.0033 i</math>
|-
| <math>10^5</math>
| <math>0.1543 + 0.1660 i</math>
| <math>0.1543 + 0.1660 i </math>
| <math>0.1543 + 0.1660 i </math>
| <math>0.1544 + 0.1663 i </math>
|-
| <math>10^6</math>
| <math>-0.1013 - 0.1887 i</math>
| <math>-0.1010 - 0.1889 i</math>
| <math>-0.1011 - 0.1890 i</math>
| <math>-0.1012 - 0.1888 i</math>
|-
| <math>10^7</math>
| <math>-0.1018 + 0.1135 i</math>
| <math>-0.1022 + 0.1133 i</math>
| <math>-0.1025 + 0.1128 i</math>
| <math>-0.0986 + 0.1163 i</math>
|}

Here some typical values of <math>C/B_0</math>, which is significantly smaller than either <math>A/B_0</math> or <math>B/B_0</math>:


{| border=1
|-
! style="text-align:left;"| <math>x</math>
! <math>C/B_0</math>
! <math>C^{eff}/B^{eff}_0</math>
|-
| <math>10^3</math>
| <math>-0.1183 + 0.0697i</math>
| <math>-0.0581 + 0.0823
i</math>
|-
| <math>10^4</math>
| <math>-0.0001 - 0.0184 i</math>
| <math>-0.0001 - 0.0172 i</math>
|-
| <math>10^5</math>
| <math>-0.0033 - 0.0005i</math>
| <math>-0.0031 - 0.0005i</math>
|-
| <math>10^6</math>
| <math>-0.0001 - 0.0006 i</math>
| <math>-0.0001 - 0.0006 i</math>
|-
| <math>10^7</math>
| <math>-0.0000 - 0.0001 i</math>
| <math>-0.0000 - 0.0001 i</math>
|}

Some values of <math>H_t</math> and its approximations at small values of <math>x</math> [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493456 source] [https://terrytao.wordpress.com/2018/03/02/polymath15-fifth-thread-finishing-off-the-test-problem/#comment-493715 source]:

{| border=1
|-
! style="text-align:left;"| <math>x</math>
! <math>H_t</math>
! <math>A+B</math>
! <math>A'+B'</math>
! <math>A^{eff}+B^{eff}</math>
! <math>A^{toy}+B^{toy}</math>
! <math>A+B-C</math>
! <math>A^{eff}+B^{eff}-C^{eff}</math>
|-
| <math>10</math>
| <math>(3.442 - 0.168 i) \times 10^{-2}</math>
| 0
| 0
| 0
| N/A
| N/A
| <math>(3.501 - 0.316 i) \times 10^{-2}</math>
|-
| <math>30</math>
| <math>(-1.000 - 0.071 i) \times 10^{-4}</math>
| <math>(-0.650 - 0.188 i) \times 10^{-4}</math>
| <math>(-0.211 - 0.192 i) \times 10^{-4}</math>
| <math>(-0.670 - 0.114 i) \times 10^{-4}</math>
| <math>(-0.136 + 0.021 i) \times 10^{-4}</math>
| <math>(-1.227 - 0.058 i) \times 10^{-4}</math>
| <math>(-1.032 - 0.066 i) \times 10^{-4}</math>
|-
| <math>100</math>
| <math>(6.702 + 3.134 i) \times 10^{-16}</math>
| <math>(2.890 + 3.667 i) \times 10^{-16}</math>
| <math>(2.338 + 3.742 i) \times 10^{-16}</math>
| <math>(2.955 + 3.650 i) \times 10^{-16}</math>
| <math>(0.959 + 0.871 i) \times 10^{-16}</math>
| <math>(6.158 + 12.226 i) \times 10^{-16}</math>
| <math>(6.763 + 3.074 i) \times 10^{-16} </math>
|-
| <math>300</math>
| <math>(-4.016 - 1.401 i) \times 10^{-49}</math>
| <math>(-5.808 - 1.140 i) \times 10^{-49}</math>
| <math>(-5.586 - 1.228 i) \times 10^{-49}</math>
| <math>(-5.824 - 1.129 i) \times 10^{-49}</math>
| <math>(-2.677 - 0.327 i) \times 10^{-49}</math>
| <math>(-3.346 + 6.818 i) \times 10^{-49}</math>
| <math>(-4.032 - 1.408 i) \times 10^{-49}</math>
|-
| <math>1000</math>
| <math>(0.015 + 3.051 i) \times 10^{-167}</math>
| <math>(-0.479 + 3.126 i) \times 10^{-167}</math>
| <math>(-0.516 + 3.135 i) \times 10^{-167}</math>
| <math>(-0.474 + 3.124 i) \times 10^{-167}</math>
| <math>(-0.406 + 2.051 i) \times 10^{-167}</math>
| <math>(0.175 + 3.306 i) \times 10^{-167}</math>
| <math>(0.017 + 3.047 i) \times 10^{-167}</math>
|-
| <math>3000</math>
| <math>(-1.144+ 1.5702 i) 10^{-507}</math>
| <math>(-1.039+ 1.5534 i) 10^{-507}</math>
| <math>(-1.039+ 1.5552 i) 10^{-507}</math>
| <math>(-1.038+ 1.5535 i) 10^{-507}</math>
| <math>(-0.925+ 1.3933 i) 10^{-507}</math>
| <math>(-1.155+ 1.5686 i) 10^{-507}</math>
| <math>(-1.144+ 1.5701 i) 10^{-507}</math>
|-
| <math>10000</math>
| <math>(-0.558 - 4.088 i) \times 10^{-1700}</math>
| <math>(-0.692 - 4.067 i) \times 10^{-1700}</math>
| <math>(-0.687 - 4.067 i) \times 10^{-1700}</math>
| <math>(-0.692 - 4.066 i) \times 10^{-1700}</math>
| <math>(-0.673 - 3.948 i) \times 10^{-1700}</math>
| <math>(-0.548 - 4.089 i) \times 10^{-1700}</math>
| <math>(-0.558 - 4.088 i) \times 10^{-1700}</math>
|-
| <math>30000</math>
| <math>(3.160 - 6.737) \times 10^{-5110}</math>
| <math>(3.065 - 6.722) \times 10^{-5100}</math>
| <math>(3.066 - 6.722) \times 10^{-5100}</math>
| <math>(3.065 - 6.722) \times 10^{-5100}</math>
| <math>(2.853 - 6.286) \times 10^{-5100}</math>
| <math>(3.170 - 6.733) \times 10^{-5100}</math>
| <math>(3.160 - 6.737) \times 10^{-5100}</math>
|}

== Controlling |A+B|/|B_0| ==

See [[Controlling A+B/B_0]].

== Controlling |H_t-A-B|/|B_0| ==

See [[Controlling H_t-A-B/B_0]].

== Small values of x ==

Tables of <math>H_t(x+iy)</math> for small values of <math>x</math>:

* [https://github.com/km-git-acc/dbn_upper_bound/tree/master/dbn_upper_bound/python/research/H_t%20at%20small%20x x=0 to x=300 with step size 0.1]
* [https://pastebin.com/fim2swFu x=200 to x=600 with step size 0.1]
* [https://pastebin.com/jvSvDP69 x=600 to x=1000 with step size 0.1]