Polymath15 test problem

We are initially focusing attention on the following

Test problem For [math]\displaystyle{ t=y=0.4 }[/math], can one prove that [math]\displaystyle{ H_t(x+iy) \neq 0 }[/math] for all [math]\displaystyle{ 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]\displaystyle{ H_t(x+iy) }[/math] at [math]\displaystyle{ y=0.4 }[/math]) that one can show that [math]\displaystyle{ H_t(x+iy) \neq 0 }[/math] for all [math]\displaystyle{ y \geq 0.4 }[/math] as well. This would give a new upper bound

[math]\displaystyle{ \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]\displaystyle{ x }[/math] we expect to be able to establish this by direct calculation of [math]\displaystyle{ H_t(x+iy) }[/math]. For medium or large values, the strategy is to use a suitable approximation

[math]\displaystyle{ H_t(x+iy) \approx A + B }[/math]

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

[math]\displaystyle{ \frac{|H_t - A - B|}{|B_0|} \lt \frac{|A + B|}{|B_0|}. }[/math]

Thus one will seek upper bounds on the error [math]\displaystyle{ \frac{|H_t - A - B|}{|B_0|} }[/math] and lower bounds on [math]\displaystyle{ \frac{|A+B|}{|B_0|} }[/math] for various ranges of [math]\displaystyle{ x }[/math]. Numerically it seems that the RHS stays above 0.4 as soon as [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ s := \frac{1-y+ix}{2} }[/math]

[math]\displaystyle{ N := \lfloor \frac{\mathrm{Im} s}{2\pi} \rfloor = \lfloor \frac{x}{4\pi} \rfloor. }[/math]

This was modified slightly to

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ s := \frac{1-y+ix}{2} }[/math]

[math]\displaystyle{ N := \lfloor \frac{\mathrm{Im} s}{2\pi} \rfloor = \lfloor \frac{x}{4\pi} \rfloor. }[/math]

In Effective bounds on H_t - second approach, a more refined approximation was introduced:

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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})} }[/math]

[math]\displaystyle{ 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]\displaystyle{ \alpha_1(s) := \frac{1}{2s} + \frac{1}{s-1} + \frac{1}{2} \log \frac{s}{2\pi} }[/math]

[math]\displaystyle{ N := \lfloor \sqrt{ \frac{T'}{2\pi}} \rfloor }[/math]

[math]\displaystyle{ T' := \frac{x}{2} - \frac{\pi t}{8}. }[/math]

Finally, a simplified approximation is

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ N := \lfloor \sqrt{\frac{x}{4\pi}} \rfloor. }[/math]

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

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

Here some typical values of [math]\displaystyle{ B/B_0 }[/math] (note that [math]\displaystyle{ B/B_0 }[/math] and [math]\displaystyle{ B'/B'_0 }[/math] are identical):

[math]\displaystyle{ x }[/math]	[math]\displaystyle{ B/B_0 }[/math]	[math]\displaystyle{ B'/B'_0 }[/math]	[math]\displaystyle{ B^{eff}/B^{eff}_0 }[/math]	[math]\displaystyle{ B^{toy}/B^{toy}_0 }[/math]
[math]\displaystyle{ 10^3 }[/math]	[math]\displaystyle{ 0.7722 + 0.6102 i }[/math]	[math]\displaystyle{ 0.7722 + 0.6102 i }[/math]	[math]\displaystyle{ 0.7733 + 0.6101 i }[/math]	[math]\displaystyle{ 0.7626 + 0.6192 i }[/math]
[math]\displaystyle{ 10^4 }[/math]	[math]\displaystyle{ 0.7434 - 0.0126 i }[/math]	[math]\displaystyle{ 0.7434 - 0.0126 i }[/math]	[math]\displaystyle{ 0.7434 - 0.0126 i }[/math]	[math]\displaystyle{ 0.7434 - 0.0124 i }[/math]
[math]\displaystyle{ 10^5 }[/math]	[math]\displaystyle{ 1.1218 - 0.3211 i }[/math]	[math]\displaystyle{ 1.1218 - 0.3211 i }[/math]	[math]\displaystyle{ 1.1218 - 0.3211 i }[/math]	[math]\displaystyle{ 1.1219 - 0.3213 i }[/math]
[math]\displaystyle{ 10^6 }[/math]	[math]\displaystyle{ 1.3956 - 0.5682 i }[/math]	[math]\displaystyle{ 1.3956 - 0.5682 i }[/math]	[math]\displaystyle{ 1.3955 - 0.5682 i }[/math]	[math]\displaystyle{ 1.3956 - 0.5683 i }[/math]
[math]\displaystyle{ 10^7 }[/math]	[math]\displaystyle{ 1.6400 + 0.0198 i }[/math]	[math]\displaystyle{ 1.6400 + 0.0198 i }[/math]	[math]\displaystyle{ 1.6401 + 0.0198 i }[/math]	[math]\displaystyle{ 1.6400 - 0.0198 i }[/math]

Here some typical values of [math]\displaystyle{ A/B_0 }[/math]:

[math]\displaystyle{ x }[/math]	[math]\displaystyle{ A/B_0 }[/math]	[math]\displaystyle{ A'/B'_0 }[/math]	[math]\displaystyle{ A^{eff}/B^{eff}_0 }[/math]	[math]\displaystyle{ A^{toy}/B^{toy}_0 }[/math]
[math]\displaystyle{ 10^3 }[/math]	[math]\displaystyle{ -0.3856 - 0.0997 i }[/math]	[math]\displaystyle{ -0.3857 - 0.0953 i }[/math]	[math]\displaystyle{ -0.3854 - 0.1002 i }[/math]	[math]\displaystyle{ -0.4036 - 0.0968 i }[/math]
[math]\displaystyle{ 10^4 }[/math]	[math]\displaystyle{ -0.2199 - 0.0034 i }[/math]	[math]\displaystyle{ -0.2199 - 0.0036 i }[/math]	[math]\displaystyle{ -0.2199 - 0.0033 i }[/math]	[math]\displaystyle{ -0.2208 - 0.0033 i }[/math]
[math]\displaystyle{ 10^5 }[/math]	[math]\displaystyle{ 0.1543 + 0.1660 i }[/math]	[math]\displaystyle{ 0.1543 + 0.1660 i }[/math]	[math]\displaystyle{ 0.1543 + 0.1660 i }[/math]	[math]\displaystyle{ 0.1544 + 0.1663 i }[/math]
[math]\displaystyle{ 10^6 }[/math]	[math]\displaystyle{ -0.1013 - 0.1887 i }[/math]	[math]\displaystyle{ -0.1010 - 0.1889 i }[/math]	[math]\displaystyle{ -0.1011 - 0.1890 i }[/math]	[math]\displaystyle{ -0.1012 - 0.1888 i }[/math]
[math]\displaystyle{ 10^7 }[/math]	[math]\displaystyle{ -0.1018 + 0.1135 i }[/math]	[math]\displaystyle{ -0.1022 + 0.1133 i }[/math]	[math]\displaystyle{ -0.1025 + 0.1128 i }[/math]	[math]\displaystyle{ -0.0986 + 0.1163 i }[/math]

Controlling |A+B|/|B_0|

Some numerical data on [math]\displaystyle{ |A+B/B_0| }[/math] source and also [math]\displaystyle{ \mathrm{Re} \frac{A+B}{B_0} }[/math] source, using a step size of 1 for [math]\displaystyle{ x }[/math], suggesting that this ratio tends to oscillate roughly between 0.5 and 3 for medium values of [math]\displaystyle{ x }[/math]:

range of [math]\displaystyle{ x }[/math]	minimum value	max value	average value	standard deviation	min real part	max real part
0-1000	0.179	4.074	1.219	0.782
4.06
1000-2000	0.352	4.403	1.164	0.712	0.02	4.43
2000-3000	0.352	4.050	1.145	0.671	0.15	3.99
3000-4000	0.338	4.174	1.134	0.640	0.34	4.48
4000-5000	0.386	4.491	1.128	0.615	0.33	4.33
5000-6000	0.377	4.327	1.120	0.599	0.377	4.327
[math]\displaystyle{ 1-10^5 }[/math]	0.179	4.491	1.077	0.455
4.48
[math]\displaystyle{ 10^5-2 \times 10^5 }[/math]	0.488	3.339	1.053	0.361	0.48	3.32
[math]\displaystyle{ 2 \times 10^5-3 \times 10^5 }[/math]	0.508	3.049	1.047	0.335	0.50	3.00
[math]\displaystyle{ 3 \times 10^5-4 \times 10^5 }[/math]	0.517	2.989	1.043	0.321	0.52	2.97
[math]\displaystyle{ 4 \times 10^5-5 \times 10^5 }[/math]	0.535	2.826	1.041	0.310	0.53	2.82
[math]\displaystyle{ 5 \times 10^5-6 \times 10^5 }[/math]	0.529	2.757	1.039	0.303	0.53	2.75
[math]\displaystyle{ 6 \times 10^5-7 \times 10^5 }[/math]	0.548	2.728	1.038	0.296	0.55	2.72

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

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