Second attempt at computing H t(x) for negative t

We are interested in approximating

[math]\displaystyle{ H_t(x) = \int_{\bf R} \frac{1}{8} \xi(\frac{1+ix}{2} + i |t|^{1/2} v) \frac{1}{\sqrt{\pi}} e^{-v^2}\ dv \quad (1.1) }[/math]

in the regime when [math]\displaystyle{ x }[/math] is large and [math]\displaystyle{ t }[/math] is large and negative.

To cancel off an exponential decay factor in the [math]\displaystyle{ \xi }[/math] function, it is convenient to shift the v variable by [math]\displaystyle{ \pi |t|^{1/2}/8 }[/math], thus

[math]\displaystyle{ H_t(x) = \frac{1}{8\sqrt{\pi}} \int_{\bf R} \xi(\frac{1+ix}{2} + i |t|^{1/2} v - \pi i |t|/8) e^{-(v - \pi |t|^{1/2}/8)^2}\ dv \quad (1.2) }[/math]

[math]\displaystyle{ = \frac{\exp( \pi^2 t / 64)}{8\sqrt{\pi}} \int_{\bf R} \xi(\frac{1+i\tilde x}{2} + i |t|^{1/2} v) e^{-v^2 + \pi |t|^{1/2} v / 4}\ dv \quad (1.3) }[/math]

where

[math]\displaystyle{ \tilde x := x - \pi |t|/4 = x + \frac{\pi t}{4}. \quad (1.4) }[/math]

Now from the definition of [math]\displaystyle{ \xi }[/math] and the Stirling approximation we have

[math]\displaystyle{ \frac{1}{8} \xi(s) \approx M_0(s) \zeta(s)\quad (1.5) }[/math]

where [math]\displaystyle{ M_0 }[/math] is defined in (6) of the writeup. Thus

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{\pi}} \int_{\bf R} M_0(\frac{1+i\tilde x}{2} + i |t|^{1/2} v) \zeta(\frac{1+i\tilde x}{2} + i |t|^{1/2} v) e^{-v^2 + \pi |t|^{1/2} v / 4}\ dv.\quad (1.6) }[/math]

By Taylor expansion we have

[math]\displaystyle{ M_0(\frac{1+i\tilde x}{2} + i |t|^{1/2} v) \approx M_0(\frac{1+i\tilde x}{2}) \exp( \alpha( \frac{1+i\tilde x}{2} ) i |t|^{1/2} v + \alpha'(\frac{1+i \tilde x}{2}) \frac{-|t| v^2}{2} )\quad (1.7) }[/math]

where [math]\displaystyle{ \alpha }[/math] is defined in equation (8) of the writeup. We have the approximations

[math]\displaystyle{ \alpha(\frac{1+i\tilde x}{2} ) \approx \frac{1}{2} \log \frac{\tilde x}{4\pi} + \frac{i\pi}{4} \quad (1.8) }[/math]

and

[math]\displaystyle{ \alpha'(\frac{1+i\tilde x}{2} ) \approx \frac{-i}{\tilde x} \quad (1.9) }[/math]

and hence

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{\pi}} M_0(\frac{1+i\tilde x}{2}) \int_{\bf R} \exp( \frac{i |t|^{1/2} v}{2} \log \frac{\tilde x}{4\pi} - \pi |t|^{1/2} v/4 + i |t| v^2 / 2\tilde x) \zeta(\frac{1+i\tilde x}{2} + i |t|^{1/2} v) e^{-v^2 + \pi |t|^{1/2} v / 4}\ dv.\quad (1.10) }[/math]

The two factors of [math]\displaystyle{ \exp( \pi |t|^{1/2} v/4 ) }[/math] cancel. If we now write

[math]\displaystyle{ N := \sqrt{\frac{\tilde x}{4\pi}}\quad (1.11) }[/math]

and

[math]\displaystyle{ u := |t|/N^2 = 4\pi |t|/\tilde x,\quad (1.12) }[/math]

we conclude that

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{\pi}} M_0(\frac{1+i\tilde x}{2}) \int_{\bf R} \exp( i |t|^{1/2} v \log N + i u v^2 / 8 \pi) \zeta(\frac{1+i\tilde x}{2} + i |t|^{1/2} v) e^{-v^2}\ dv.\quad (1.13) }[/math]

If we formally write [math]\displaystyle{ \zeta(s) = \sum_n \frac{1}{n^s} }[/math] (ignoring convergence issues) we obtain

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{\pi}} M_0(\frac{1+i\tilde x}{2}) \sum_n \int_{\bf R} \exp( i |t|^{1/2} v \log N + i u v^2 / 8 \pi) n^{-\frac{1+i\tilde x}{2} - i |t|^{1/2} v} e^{-v^2}\ dv\quad (1.14) }[/math]

[math]\displaystyle{ \approx \frac{\exp( \pi^2 t / 64)}{8\sqrt{\pi}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} \sum_n \int_{\bf R} \exp( - i |t|^{1/2} v \log \frac{n}{N} + i u v^2 / 8 \pi -\frac{1+i\tilde x}{2} \log \frac{n}{N} ) e^{-v^2}\ dv\quad (1.15) }[/math]

We can compute the [math]\displaystyle{ v }[/math] integral to obtain

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} \sum_n \exp( - \frac{|t| \log^2 \frac{n}{N}}{4 (1 - iu / 8 \pi)} -\frac{1+i\tilde x}{2} \log \frac{n}{N}).\quad (1.16) }[/math]

We approximate [math]\displaystyle{ \frac{1+i\tilde x}{2} \approx \frac{i\tilde x}{2} }[/math] to obtain

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} \sum_n \exp( - \frac{|t| \log^2 \frac{n}{N}}{4 (1 - iu / 8 \pi)} -\frac{i\tilde x}{2} \log \frac{n}{N})\quad (1.17) }[/math]

and then we approximate [math]\displaystyle{ \log^2 \frac{n}{N} \approx \frac{(n-N)^2}{N^2} }[/math] and [math]\displaystyle{ |t| = u N^2 }[/math] and [math]\displaystyle{ \tilde x= 4\pi N^2 }[/math] to obtain

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} \sum_n \exp( - \frac{u(n-N)^2}{4 (1 - iu / 8 \pi)} - 2\pi i N^2 \log \frac{n}{N}).\quad (1.18) }[/math]

Next, we use the Taylor approximation

[math]\displaystyle{ \log \frac{n}{N} \approx \frac{n-N}{N} - \frac{(n-N)^2}{2N^2} + \frac{(n-N)^3}{3N^3}\quad (1.19) }[/math]

to obtain

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} \sum_n \exp( - \frac{u(n-N)^2}{4 (1 - iu / 8 \pi)} - 2\pi i N(n-N) + \pi i (n-N)^2 - \frac{2\pi i}{3N} (n-N)^3).\quad (1.20) }[/math]

Writing

[math]\displaystyle{ N(n-N) = \frac{1}{2} n^2 - \frac{1}{2} N^2 - \frac{1}{2} (N-n)^2 }[/math]

this becomes

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} e^{\pi i N^2} \sum_n \exp( - \frac{u(n-N)^2}{4 (1 - iu / 8 \pi)} - \pi i n^2 + 2 \pi i (n-N)^2 - \frac{2\pi i}{3N} (n-N)^3)\quad (1.23) }[/math]

[math]\displaystyle{ \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} e^{\pi i N^2} \sum_n \exp( \frac{2\pi i(n-N)^2}{(1 - iu / 8 \pi)} + \pi i n - \frac{2\pi i}{3N} (n-N)^3)\quad (1.24) }[/math]

By Poisson summation, this is

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} e^{\pi i N^2} \sum_{m \in {\bf Z}} \int_{\bf R} \exp( \frac{2\pi i(x-N)^2}{(1 - iu / 8 \pi)} + 2\pi i m x + \pi i x - \frac{2\pi i}{3N} (x-N)^3)\ dx\quad (1.25) }[/math]

[math]\displaystyle{ \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} e^{\pi i N^2} \sum_{m \in {\bf Z} + \frac{1}{2}} \int_{\bf R} \exp( \frac{2\pi i(x-N)^2}{1 - iu / 8 \pi} + 2\pi i m x - \frac{2\pi i}{3N} (x-N)^3)\ dx\quad (1.26) }[/math]

[math]\displaystyle{ \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} e^{\pi i N^2} \sum_{m \in {\bf Z} + \frac{1}{2}} e^{2\pi i m N} \int_{\bf R} \exp( - \frac{2\pi ix^2}{1 - iu / 8 \pi} + 2\pi i m x - \frac{2\pi i}{3N} x^3)\ dx\quad (1.27) }[/math]

The integral here can be evaluated as an Airy integral, but perhaps a Taylor expansion of the last term is a better approach?

We return to (1.18) and try to apply Poisson summation without using the Taylor expansion. We first observe that as [math]\displaystyle{ \exp( \pi i n^2 - \pi i n ) = 1 }[/math] for all integers [math]\displaystyle{ n }[/math], we have

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} \sum_n \exp( - \frac{u(n-N)^2}{4 (1 - iu / 8 \pi)} + \pi i n^2 - \pi i n - 2\pi i N^2 \log \frac{n}{N}).\quad (1.28) }[/math]

(this is to try to reduce the oscillation of the phase near [math]\displaystyle{ n \sim N }[/math]). By Poisson summation (extending the range of n to the entire integers), this is

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} \sum_{m \in {\bf Z}} \int_{\bf R} \exp( - \frac{u(X-N)^2}{4 (1 - iu / 8 \pi)} + \pi i X^2 - \pi i X + 2\pi i m X - 2\pi i N^2 \log \frac{X}{N})\ dX.\quad (1.29) }[/math]

(using [math]\displaystyle{ X }[/math] as the variable of integration to distinguish it from the initial variable [math]\displaystyle{ x }[/math]. Shifting [math]\displaystyle{ m }[/math] by 1/2, this is

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}}\exp(\pi i N^2) \sum_{m \in {\bf Z} + \frac{1}{2}} f_m( u, N ) \quad (1.30) }[/math]

where

[math]\displaystyle{ f_m(u,N) := \int_{\bf R} \exp( - \frac{u(X-N)^2}{4 (1 - iu / 8 \pi)} + \pi i X^2 - \pi i N^2 + 2\pi i m X - 2\pi i N^2 \log \frac{X}{N})\ dX.\quad (1.31) }[/math]

These functions should be roughly 1-periodic in N and decay quickly as [math]\displaystyle{ |m| \to \infty }[/math]; they are roughly like the m^th Fourier harmonics of [math]\displaystyle{ H_t }[/math] in the N variable.

Shifting [math]\displaystyle{ X }[/math] by [math]\displaystyle{ N }[/math], we have

[math]\displaystyle{ f_m(u,N) = \int_{\bf R} \exp( - \frac{uX^2}{4 (1 - iu / 8 \pi)} + \pi i (X+N)^2 - \pi i N^2 + 2\pi i m (X+N) - 2\pi i N^2 \log(1+\frac{X}{N}) )\ dX\quad (1.32) }[/math]

[math]\displaystyle{ = \exp( 2\pi i m N) \int_{\bf R} \exp( \frac{2\pi i}{1-\frac{iu}{8\pi}} X^2 + 2\pi i m X - 2\pi i N^2 [\log(1+\frac{X}{N}) - \frac{X}{N} + \frac{X^2}{2N^2}] )\ dX. (1.33) }[/math]

If we ignore the term [math]\displaystyle{ 2\pi i N^2 [\log(1+\frac{X}{N}) - \frac{X}{N} + \frac{X^2}{2N^2}] = O( X^3 / N ) }[/math], we would have

[math]\displaystyle{ f_m(u,N) \approx \exp( 2\pi i m N) \int_{\bf R} \exp( \frac{2\pi i}{1-\frac{iu}{8\pi}} X^2 + 2\pi i m X)\ dX (1.34) }[/math]

[math]\displaystyle{ = \sqrt{\pi i (1-\frac{iu}{8\pi})} \exp( 2\pi i m N - \frac{\pi i m^2}{2} (1 - \frac{iu}{8\pi}) )(1.35) }[/math]

hence

[math]\displaystyle{ H_t(x) \approx \exp( \pi^2 t / 64) \sqrt{\pi i} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}}\exp(\pi i N^2) \sum_{m \in {\bf Z} + \frac{1}{2}} \exp( 2\pi i m N - \frac{\pi i m^2}{2} (1 - \frac{iu}{8\pi}) ) \quad (1.36) }[/math]

[math]\displaystyle{ = \exp( \pi^2 t / 64) \sqrt{\pi i} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}}\exp(\pi i N^2) \sum_{m \in {\bf Z} + \frac{1}{2}} e^{-m^2 u / 16} \exp( 2\pi i m N - \frac{\pi i m^2}{2} ) \quad (1.37) }[/math]

[math]\displaystyle{ = \exp( \pi^2 t / 64) \sqrt{\pi i} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}}\exp(\pi i N^2) \sum_{m \in {\bf Z} + \frac{1}{2}} e^{-m^2 u / 16} \cos( 2\pi m N ) \exp( - \frac{\pi i m^2}{2} ) \quad (1.38) }[/math]

[math]\displaystyle{ = 2\exp( \pi^2 t / 64 + \frac{\pi i}{8} - u/64) \sqrt{\pi} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}}\exp(\pi i N^2) \sum_{n=0}^\infty e^{-n(n+1) u / 16} \cos( 2\pi (n+\frac{1}{2}) N ) \exp( - \frac{\pi i n(n+1)}{2} ) \quad (1.39) }[/math]

This seems to be an adequate approximation for large [math]\displaystyle{ u }[/math], but not for small [math]\displaystyle{ u }[/math].

We return now to (1.20) and write it slightly differently. We compute

[math]\displaystyle{ \exp( - 2\pi i N(n-N) + \pi i (n-N)^2 ) = \exp( \pi i n^2 - 4\pi i N n + 3 \pi i N^2 ) }[/math]

[math]\displaystyle{ = \exp( -\pi i n - 4\pi i N n + 3 \pi i N^2 ) }[/math]

[math]\displaystyle{ = \exp(- 2\pi i \{ 2N + \frac{1}{2} \} n + 3 \pi i N^2 ) }[/math]

where [math]\displaystyle{ \{x\} }[/math] is the fractional part. We thus have

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} e^{3\pi i N^2} \sum_n \exp( - \frac{u(n-N)^2}{4 (1 - iu / 8 \pi)} - 2\pi i n \{2N + \frac{1}{2}\} - \frac{2\pi i}{3N} (n-N)^3).\quad (1.40) }[/math]

When [math]\displaystyle{ u }[/math] is small, the summands here are fairly slowly varying, which suggests the use of Poisson summation:

[math]\displaystyle{ H_t(x) \approx \frac{\exp( \pi^2 t / 64)}{\sqrt{1-\frac{iu}{8\pi}}} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} e^{3\pi i N^2} \sum_{m \in {\bf Z}} g_m(N,u) \quad (1.41) }[/math]

where

[math]\displaystyle{ g_m(N,u) = \int_{\bf R} \exp( - \frac{u(X-N)^2}{4 (1 - iu / 8 \pi)} + 2\pi i X (m - \{2N + \frac{1}{2}\}) - \frac{2\pi i}{3N} (X-N)^3)\ dX.\quad (1.42) }[/math]

The region of integration here should contain a neighbourhood of [math]\displaystyle{ N }[/math] as this is where the integrand is largest. We can shift [math]\displaystyle{ X }[/math] by [math]\displaystyle{ N }[/math] to obtain

[math]\displaystyle{ g_m(N,u) = e^{2\pi i N (m - \{2N + \frac{1}{2}\})} \int_{\bf R} \exp( - \frac{uX^2}{4 (1 - iu / 8 \pi)} + 2\pi i X (m - \{2N + \frac{1}{2}\}) - \frac{2\pi i}{3N} X^3)\ dX.\quad (1.43) }[/math]

and now the region of integration should be centered at [math]\displaystyle{ X=0 }[/math].

Suppose we neglect the third order term [math]\displaystyle{ \frac{2\pi i}{3N} X^3 }[/math]. Then

[math]\displaystyle{ g_m(N,u) \approx e^{2\pi i N (m - \{2N + \frac{1}{2}\})} \int_{\bf R} \exp( - \frac{uX^2}{4 (1 - iu / 8 \pi)} + 2\pi i X (m - \{2N + \frac{1}{2}\})\ dX\quad (1.44) }[/math]

[math]\displaystyle{ \approx \sqrt{\frac{4 \pi (1-iu/8\pi)}{u}} e^{2\pi i N (m - \{2N + \frac{1}{2}\})} \exp( -\frac{4 \pi^2 (m - \{2N + \frac{1}{2}\})^2 (1 - iu/8\pi) }{u} ) \quad (1.45) }[/math]

For small [math]\displaystyle{ u }[/math], this decays quickly for [math]\displaystyle{ m }[/math] far from [math]\displaystyle{ \{2N + \frac{1}{2}\} }[/math]. For [math]\displaystyle{ N }[/math] close to an integer or half-integer, [math]\displaystyle{ \{2N+\frac{1}{2}\} }[/math] is close to 1/2, and so the m=0 and m=1 terms should be dominant.

Let's compute the integral in (1.43). We have

[math]\displaystyle{ g_m(N,u) = \exp( i N c ) \int_{\bf R} \exp( -\frac{ia}{3} X^3 + i b X^2 + i c X )\ dX \quad (1.46) }[/math]

where

[math]\displaystyle{ a := \frac{2\pi}{N} \quad (1.47) }[/math]

[math]\displaystyle{ b := i \frac{u}{4(1-iu/8\pi)} \quad (1.48) }[/math]

[math]\displaystyle{ c := 2 \pi (m - \{ 2N + \frac{1}{2} \} ). \quad (1.49) }[/math]

Change variables [math]\displaystyle{ X = a^{-1/3} Y }[/math]:

[math]\displaystyle{ g_m(N,u) = a^{-1/3} \exp( i N c ) \int_{\bf R} \exp( -\frac{i}{3} Y^3 + i b a^{-2/3} Y^2 + i c a^{-1/3} Y )\ dY \quad (1.50) }[/math]

Change variables [math]\displaystyle{ Y = Z + b a^{-2/3} }[/math] and shift the contour of integration:

[math]\displaystyle{ g_m(N,u) = a^{-1/3} \exp( i N c + 4 i b^3 a^{-2}/3 + i bc a^{-1}) \int_{\bf R} \exp( -\frac{i}{3} Z^3 + i(c a^{-1/3} + b^2 a^{-4/3}) Z )\ dZ.\quad (1.51) }[/math]

Since

[math]\displaystyle{ \mathrm{Ai}(x) = \frac{1}{\pi} \int_0^\infty \cos(\frac{t^3}{3} + xt)\ dt \quad (1.52) }[/math]

[math]\displaystyle{ = \frac{1}{2\pi} \int_{\bf R} \exp(i\frac{t^3}{3} + ixt)\ dt \quad (1.53) }[/math]

[math]\displaystyle{ = \frac{1}{2\pi} \int_{\bf R} \exp(-i\frac{Z^3}{3} - ixZ)\ dZ \quad (1.54) }[/math]

we thus conclude that

[math]\displaystyle{ g_m(N,u) = 2\pi a^{-1/3} \exp( i N c + 4 i b^3 a^{-2}/3 + i bc a^{-1}) \mathrm{Ai}( - (c a^{-1/3} + b^2 a^{-4/3}) ).\quad (1.55) }[/math]

Note: b is complex valued, so the argument of the Airy function is also complex. Hopefully this is not going to be an issue. Also as N is large, the argument is going to be fairly large. But this could be a good thing; it means that we could hope to use one of the known asymptotics of the Airy function to get a more tractable approximation.