Prove that $ \sum_{k=1}^{\infty} (-1)^k q^{k^2} \geq \frac{2}{\pi}\ \sum_{k=1}^{\infty} \frac{(-1)^k}{2k-1}{q}^{\frac{ (2k-1)^2}{4}}  $ for $0<q<1$ (updated bounty)

The title explains it. Prove that :

$ \sum_{k=1}^{\infty} {(-1)^k\rm e}^{-\hat{t} k^2} \geq \frac{2}{\pi}\ \sum_{k=1}^{\infty} \frac{(-1)^k}{2k-1}{\rm e}^{\frac{-\hat{t} (2k-1)^2}{4}}  $ for $\hat{t}>0$

For some background on the problem, you can see my recent paper: 

https://doi.org/10.3390/bios13090845

I am an engineer and not a mathematician. However, it bugged me (and a reviewer of the paper!) that I wasn't able to prove that Equation 13 is always greater than Equation 15.

Graphically (Figure 5), it seems to be true that the flux sensor result $\bar{m}_{Flux}\geq\bar{m}_{Conc}$ for a given $\bar{t}$, but I was not able to prove it analytically! Would love to know how it's done. I imagine it will be something related to the fact that exponentials are always greater than zero. Maybe some clever stuff about the limit as $\bar{t}\to 0$ and taking the derivative? The difference function $\bar{m}_{Flux}-\bar{m}_{Conc}$ seems to be a single-peaked function. I'm not sure if that helps!

Note, in the paper I use $\bar{t} = \frac{t D}{L^2}$ whereas above I have simplified it further and use $\hat{t} = \frac{t D \pi^2}{ L^2}$

you can substitute ${\rm e}^{-t} = q$, for $t>0$, or $0<q<1$ (as I did in title of post vs body of text)

Note also that the left hand side is a Jacobi elliptic functions, and the right hand side sort of looks like an expansion of ArcTan. I'm not sure if that helps at all...

I recently was awarded $10 for answering a question correctly, so I have put that towards the bounty!

  • Erdos Erdos
    0

    Can you compare D with L? Could we assume that D is greater that L? The result should be true under this assumption, but if D is significantly smaller that L this may not be true. Is that compatible with your intuition about this problem?

    • D is the diffusion coefficient (units of cm^2/s) and L is a length (units of cm). So your question doesn't make sense. I non-dimensionalized the problem using $\bar{t} = t D/L^2$ in the paper, and $\bar{t} = t D/(\Pi^2 L^2)$ above. So if I translate what you're saying, it's basically, "is it true when \bar{t} is large and small?" and the answer is "yes I believe it is true for all $\bar{t}>0$".

  • Mathe Mathe
    +1

    I believe I have solved the problem! I had already spent a good chunk of time on it, when you shared this problem last time. Would you consider increasing the bounty to $50?

    • Mathe Mathe
      +1

      Oh, nevermind, I made a mistake. It's not done yet.

  • Sounds promising! Thank you for spending your time on this problem. I have also spent a lot of time on it!

  • M F H M F H
    0

    you should avoid using index "i" (also "l" = lowercase L) and prefer other choices (k, m, n, ...), especially in the exponential most mathematicians will always see "i" as the imaginary unit at a first glance ...

    • thank you and good point. updated from i to k. too bad l didn't change it in time for paper to be published! and yes i just used lowercase L there to be punny :)

  • M F H M F H
    0

    We can rewrite the inequality as $ \pi \sum (-1)^k \, q^k^2 > \sum (-1)^k / (k-1/2) \, q^(k-1/2)^2 $, where q = exp(-t) is an arbitrary number between 0 and 1. The L.H.S. can be written in terms of the elliptic theta function/constant $ ( \pi / 2 ) ( \theta_4 (q) - 1 ) $. (That's more or less the definition: \theta_4 (q) := 1+ 2 sum (-1)^k q^kΒ².) Maybe a possible approach is to check what the first few terms look like, and approximate/estimate later terms (k > 5 ?) by dropping the 1/2 ?

    • yes LHS can be easily rewritten as elliptic theta. how does that relate to RHS though? when you say "terms" are you referring to some series expansion of elliptic theta that ends up looking like RHS? i like the q=exp(-t) substitution though! maybe a step in the right direction

  • M F H M F H
    0

    No I meant the first terms of the sum, i.e. what you get for k = 1,2,3,4, say. Given that you have kΒ² in the exponent of q9<1, later terms are very likely "negligible" or more precisely very easily bounded from above.

  • M F H M F H
    0

    I think one might try to separate even and odd indices. The odd indices 2k-1 from the LHS might be rearranged with terms from the RHS also having exponents (2k-1)... I didn't yet have the time to develop that.

Answer

With the simplifications I suggested, in particular writing the left hand side (LHS) as 
$f(q) = \sum_{k\ge1} (-1)^k q^{k^2}$
and similar for the R.H.S. of the inequality,
$g(q) = \frac2\pi\sum_{k\ge1}\frac{(-1)^k}{2k-1}\,q^{(k-1/2)^2}$, 
the question amounts to show that the function
$h(q) := f(q) - g(q)$
is nonnegative over the open interval (0, 1).
(As we shall see, this function admits the limit zero at both endpoints of that interval and could thus be extended by continuity to the closed interval.)

For a physicist or engineer it could be proof enough to plot that function on the interval (0,1) ; that plot looks as follows :



Here is code that can be pasted into the free PARI/GP online interpreter at http://pari.math.u-bordeaux.fr/gp.html to reproduce this plot or a "zoom" on any subinterval $[a,b] \subset (0, 1)$:

{ ( f(q) = sumalt(k=1,(-q)^k^2) );
( g(q) = sumalt(k=1, (-1)^k/(2*k-1)*q^(k-.5)^2)*2/Pi );
ploth( q=1e-5, 1-1e-5, f(q)-g(q)) }

The plot shows (upon "zooming in" where necessary, e.g., q=0.08, 0.09) that:

(1) $h(q) \to 0^+$ as $q\to 0^+$,

(2) $h(q)$ is increasing on some interval $(0,m)$, $0 < m < 1$,

(3) $h(q)$ reaches its maximum $ M = h(m) \approx 0.257972 $ at $ m \approx  0.0840863 $,

(4) $h(q)$ decreases on $(m, 1)$,

(5) $h(q) \to 0^+$ as $q\to 1^-$ (i.e., as q approaches 1 from the left),

(6) the maximum $M=h(m)$ is the only local extremum, and we have the two global minima $h=0$ at the endpoints $q=0$ and $q=1$, assuming that we extend the function by continuity to the closed interval [0, 1].

Here, (2) is a consequence from (1) (with so far unspecified $m$), and similarly (4) is a consequence from (5) if we take $m$ again unspecified. The nontrivial part of these two is that the $m$ in (2) and the one in (4) are both equal to that given in (3), i.e., the unique point where $h$ takes a local extrema which is a maximum.

The result of (3) is easily proved numerically. We will try to address further below the much more challenging question to get some formula or estimate for the numerical value of $m$.

Although a numerical approach is acceptable and often the only solution if analytical results are infeasible, this is not very satisfying for a mathematician, so I will try to establish some more analytical / algebraical results.

To that end, let us group together the terms from the two sums, and write

$ h(q) = \sum_{k\ge1} (-1)^k \, [q^{k^2} - \frac{2/\pi}{2k-1}\, q^{(k-1/2)^2}]$.    (7)

=== Behaviour near zero ===

To get a better idea, let us write out the sum of the first terms with $1 \le k\le 3$:

$ h(q) = -q + q^4 - q^9 + \frac2\pi \, ( q^{1/4} - q^{9/4} +  q^{25/4}) + O(q^{49/4})$  ( as $q\to 0^+$ ).     (8)

Obviously, for very small $q\to 0^+$, all terms are negligible w.r.t. the first term ($k=1$) in $g(q)$,

$h(q) \sim \frac 2\pi \, q^{1/4} ~$  as  $~ q\to 0^+$.

(We can also double-check this result numerically by computing $h(10^{-4}) \approx 0.06356 \approx \frac2\pi\times 0.09984 \approx \frac2\pi \,10^{-1}$.)

We also note that the exponents grow very fast (namely, like $k^2$), which means that for small $q$, the terms with larger $k$ are quickly negligible: for example, 

$h(0.1) = - 0.1 + 10^{-4} - 10^{-9} +- … +\frac2\pi \,( 10^{-1/4} - 10^{-2.25} + 10^{-6.25} -+...)$
$\approx \frac2\pi \,10^{-1/4} - 0.1 = 0.2579976 $ within less than 0.01, considering just two terms.

This is actually a quite good approximation of the value of the maximum $M \approx 0.25797$ which we have numerically located at $ q = m \approx 0.084 $.
This suggests that over the interval (0, 0.1) and presumably somewhere beyond, the function 

$h_1(q) = \frac2\pi \,q^{1/4} - q ~$  is a quite acceptable approximation of our target function $h(q)$. 

This function has its maximum where its derivative,  $h_1'(q) = \frac1{2\pi}\,q^{-3/4} - 1$, has its zero, which is at $q_1 = (2\pi)^{-3/4} \approx 0.86$. This is a first formula for an approximation of the value $m$ that appears in properties (2), (3) and (4).

We get an even better approximation if we include one or two more terms, viz

$h_2(q) = \frac2\pi \,(q^{1/4} - \frac13 \,q^{9/4}) - q ~$  or  $~ h_3(q) = \frac2\pi \,(q^{1/4} - \frac13 \,q^{9/4}) - q + q^4 $.

We find $h_2'(q) = \frac1{2\pi} (q^{-3/4} - 3\,q^{5/4}) - 1$ which has its zero at $q_2  \approx 0.0838341$, an even better approximation of the value of $m$. (There is no exact expression for this value or the subsequent approximation below: they are roots of polynomials of degree 8 and higher.)

Similarly, $~ h_3'(q) = \frac1{2\pi} (q^{-3/4} - 3\,q^{5/4}) - 1 + 4\,q^3 ~$ has its zero at $~q_3 = 0.0840860756…~$, which is an extremely good approximation of the point $m$ where $h$ takes its maximum.

[PARI code: solve(q=0.05,0.1, (q^(-3/4) - 3*q^(5/4))/2/Pi - 1 + 4*q^3) ]

=== Behaviour in $(m,1)$ ===

The function $h_3$ is a quite good approximation of $h(q)$ way beyond the maximum at $m\approx 0.1$.
It is well known that the error made by truncating an alternating series $S = \sum (-1)^k a_k$ ($a_k$ decreasing to 0) at some index $k$, is majorated by the next term, $|S - S_k| \le |a_{k+1}|$ (where $S_k$ is the partial sum up to $k$).

For example, at $q = 0.5$, the sum of all terms with $k\ge3$ in $f$ and $g$, respectively, are smaller than $0.5^{9} = \frac1{512}$ and $\frac2{5\pi}\cdot 0.5^{25/4}\sim 0.00167$, respectively (in absolute value).
(The actual error is significantly smaller, because these two major contributions to the error are of comparable size but opposite sign and so they cancel out each other to a large extend.)

We find that up to at least $q = 0.6$, the function $h_3(q)$ is larger than the sum of the two "error terms" $q^9+\frac2{5\pi}\,q^{25/4}$, which proves that the function $h$ is positive up to there at least.

In the available time, I did not find a satisfying way to prove analytically that $h$ remains positive as $q\to 1^+$, so I have nothing better to offer than a numerical proof that this is the case, considering the function on several intervals from 0.6 to 1.0.

As $q$ grows from 0.6 to 1, I will use that the infinite sum is strictly positive whenever we have some $n$ such that the partial sum
$S_n :=\sum_{k=1}^n(-1)^k(a_k-b_k) > E_{n+1}$ ,
where $E_n := |a_n| + |b_n|$, $a_k=q^{k^2}$, and $b_k=q^{(k-1/2)^2}/(\pi(k-1/2))$.

We will increase q in sufficiently small steps so that we can have confidence that the behaviour in subsequent values is "continuous". (Of course one could check further intermediate values.)

Here's the list of terms (k, $a_k$, $b_k$, $S_k$) for $q = 0.6$:

k = 1: -0.6                +0.560297     -0.0397035
k = 2: +0.1296         -0.0672356    +0.0226609
k = 3: -0.0100777    +0.00522824  +0.0178115
k = 4: +0.0002821   -0.000174235 +0.0179194
k = 5: -2.84303e-6   +2.27615e-6  +0.0179188
k = 6: +1.03144e-8  -1.12606e-8   +0.0179188

Here, as we said, the terms for k=1 and 2 sum up to $S_2 \approx 0.02266$, while those for k=3 sum up to $E_3=0.015$ in absolute value, which proves that the function is strictly positive near that point.
(Of course this remains true if we truncate only at k = 3, where the partial sum is 30 times larger than the sum of absolute values of terms for k = 4.)

List of terms (k, $a_k$, $b_k$, $S_k$) for q = 0.7:

k = 1: -0.7                 +0.582311       -0.117689
k = 2: +0.2401           -0.0951107      +0.0272998
k = 3: -0.0403536      +0.0137016      +0.000647841
k = 4: +0.00332329    -0.00115142    +0.00281972
k = 5: -0.000134107   +5.16265e-05  +0.00273724
k = 6: +2.65173e-06   -1.19317e-06  +0.00273869
k = 7: -2.56924e-08   +1.39743e-08  +0.00273868
k = 8: +1.21976e-10   -8.21398e-11  +0.00273868

Here we have to truncate at k = 4 (or later), where the partial sum is $S_4 \approx$ 2.8e-3,
and the "error terms" are $E_5$ = 1.3e-4 + 5e-5 = 1.8e-4.

List of terms (k, $a_k$, $b_k$, $S_k$) for q = 0.8:

k = 1: -0.8               +0.602078      -0.197922
k = 2: +0.4096         -0.128443      +0.0832345
k = 3: -0.134218       +0.0315662    -0.019417
k = 4: +0.0281475    -0.00591064   +0.00281989
k = 5: -0.00377789   +0.000771276 -0.000186727
k = 6: +0.000324519 -6.77578e-05 +7.00336e-05
k = 7: -1.78406e-05  +3.93993e-06 +5.61329e-05
k = 8: +6.2771e-07   -1.50176e-07  +5.66104e-05

Here we must truncate not earlier than k = 6 to have a partial sum $S_6 \approx$ 7e-5 
(significantly) larger than the sum of error terms, $E_7 \approx$ 1.8e-5 + 3.9e-6 = 2.8e-5.


Terms (k, $a_k$, $b_k$, $S_k$) for q = 0.9:

k = 1:   -0.9             +0.62007           -0.27993
k = 2:  +0.6561        -0.167419          +0.208751
k = 3:  -0.38742       +0.0659061        -0.112763
k = 4:  +0.185302     -0.025018          +0.0475208
k = 5:  -0.0717898     +0.00837623     -0.0158928
k = 6:  +0.0225284     -0.00238959     +0.00424601
k = 7:  -0.00572642    +0.000571061   -0.000909341
k = 8:  +0.00117902   -0.000113222    +0.000156456
k = 9:  -0.000196627  +1.8512e-05     -2.16593e-05
k = 10: +2.65614e-5  -2.48607e-06     +2.41599e-06
k = 11: -2.90632e-6   +2.73462e-07    -2.16872e-07
k = 12: +2.57585e-7  -2.45881e-08     +1.61249e-08
k = 13: -1.8492e-8     +1.8044e-09     -5.62678e-10
k = 14: +1.07531e-9  -1.07948e-10     +4.0468e-10
k = 15: -5.06483e-11  +5.25981e-12   +3.59291e-10

Here we cannot truncate before k=14 if we want to use the same criterion.


Terms (k, $a_k$, $b_k$, $S_k$) for q = 0.95:
k = 1:   -0.95          +0.628508    -0.321492
k = 2:  +0.814506   -0.189076     +0.303938
k = 3:  -0.630249    +0.0924023   -0.233909
k = 4:  +0.440127   -0.0485173    +0.157701
k = 5:  -0.27739      +0.0250346  -0.0946544
(...)
k = 25: -1.194685e-14    +5.524311e-16   -7.806533e-16
k = 26: +8.732874e-16    -4.083986e-17   +5.179421e-17
k = 27: -5.761138e-17    +2.729016e-18   -3.088155e-18
k = 28: +3.430098e-18    -1.648139e-19   +1.771297e-19
k = 29: -1.843113e-19    +8.995052e-21   +1.813372e-21
k = 30: +8.938091e-21    -4.436042e-22   +1.030786e-20
k = 31: -3.911872e-22    +1.976670e-23   +9.936438e-21
k = 32: +1.545154e-23    -7.957688e-25   +9.951094e-21

Here we have $S(k) > E(k+1)$ only for $k$ ≥ 30.
(PARI code to produce that table for any q is given in the Appendix.
Note that Python with standard precision of 16 digits is no more adequate at this point.)

For $q=0.98$, we find that we can truncate only at $k = 76$, even at $k = 75$ we still have one last negative partial sum:
(...)
k = 74: +8.994779 e-49   -1.728344 e-50   +4.161561 e-50
k = 75: -4.432776 e-50   +8.574730 e-52   -1.854677 e-51
k = 76: +2.098037 e-51  -4.086403 e-53    +2.024966 e-52
k = 77: -9.536802 e-53   +1.870640 e-54   +1.089992 e-52

(Obviously a precision of at least 55 digits is required to get significant results in this computation.)

In the appendix I give a program ("tab(q,...)") for computing this table for any desired value of $q$.

It would be desirable to have an expression for the rank from which on we have this property, but I found some flaws in previous attempts and finally cannot provide such an expression within the alloted time.
Several other attemps (series in $q = 1-\varepsilon$ or $q = e^{-\varepsilon}$, ...) also did not bring more enlightment so far.

===============================
=====     Appendix :    =====

=== Further sample PARI code and additional remarks and formulae ===

(f(1e-4)-g(1e-4))*Pi/2  \\ gives: 0.0998429200... , illustrating lim (f-g) ~ $\frac \pi2 \sqrt q $ near $q=0$.

f(.999)-g(.999)  \\ gives: 1.103774179 E-26

/* illustrating $ \lim f = \lim g = - 0.5$ as $q\to 1$ : */

f(0.99) \\ gives:  -0.499999999999

g(0.99) \\ gives: -0.5000000000  

Side note/insider joke: Euler would certainly have been happy that we get $\lim_{q\to1-} f(q) = -1 +1 -1 +1 -+ ... = -1/2 $ ...!
(Of course the middle expression is not to be taken serious!
But we know that the series is well convergent since it is an alternating series: signs alternate and the absolute values go to zero as increasingly high powers of a number less than 1.
The series is even absolutely convergent since, taking absolute values (which gives $f(-q)$), we have something much smaller than the geometric series $\sum _{n\ge0} q^n = \frac1{1-q} <\infty$, since only the terms with square indices $n = 1, 4, 9, 25, ...$ appear in our sum $f(-q)$.)


PARI code to produce the table (k, ak, bk, Sk):

/* Use e.g., tab(0.9). The second, optional argument is the upper limit of rows to display. By default this is set to 50. If more rows are required / desired, use e.g. tab(0.98, 99).
Also, use " default(realprecision, N) " (or the shortcut \p N) to increase the precision to N digits if the default precision (N=39 on 64 machines) is no more sufficient: 10^(-N) should always be significantly smaller than all of the terms in the last displayed row.
*/

tab(q,LIM=50)={my(S,T); for(k=1,LIM, my(A=(-q)^k^2, B=-(-1)^k*q^((k-.5)^2)/Pi/(k-1/2));
if(abs(A)+abs(B) < S, T++, T=0);
printf("k = %d:\t %+7.7g \t %+7.7g \t %+7.7g\n", k, A, B, S += A+B);
T>1 && break )}


It may also be interesting to rewrite, using $(k-1/2)^2 = k^2-k-1/4$:

$ h(q) = \sum_{k\ge1} (-1)^k \, [q^{k+1/4} - \frac{2/\pi}{2k-1}]\, q^{(k-1/2)^2}]$.   
(...)

  • M F H M F H
    0

    I put what I have so far, I think you might like to have a first answer, even if it is not yet fully satisfying concerning the mathematical / algebraic proofs. I will elaborate on the answer ; I understand that I still have about 40 days ...

  • I think your logic is solid, regarding points 1-5. I am not sure why point 6 is needed, it seems redundant. 1) OK, you have shown this to be true. 2) no proof that dh/dq>0 on (0,m) 3) lack of analytical expression for mβ‰ˆ 0.0840863 - finding this root (e.g. analytical expression m such that dh/dq = 0 .. may be difficult) 4) no proof that dh/dq<0 on (m,1) 5) not complete I see "Please review the answer, and accept or dispute it within 7 days. Otherwise, the answer will be accepted automatically."

    • M F H M F H
      0

      Yes indeed 6 is a consequence of 2 and 4 with the same m. What is important is that there is no negative minimum somewhere else between m and 1.

  • M F H M F H
    +1

    I will try to provide more about the derivative h', maybe an expression for the zero (which is m) or at least show that there is only one, which would settle the question.

    • M F H M F H
      0

      Right, although the deadline is in 40 days you have only 7 days to accept the answer, I didn't consider that... πŸ˜“Maybe that countdown is reset when I make an edit, we'll see.🀞

  • I plan to dispute it on Tuesday if there is no answer by then. Happy to provide additional feedback in the meantime.

    • M F H M F H
      0

      Hello, please leave me at least 1 more day (we are not in the same time zone so Tuesday might end for you when it only begins for me ; I have been extremely busy with teaching and administrative duties and have not found the time to complete as I desired and to address the questions you raise. I might succeed in making some minor edits this evening but I plan to do more and better tomorrow afternoon (UTC - 400), in the morning I have some more lectures to give, sorry...

  • I will dispute it in 24 hours, I think that's around UTC 3am Wednesday. Best of luck!

    • Hi Bob, MFH has clearly spent several hours to answer your question. I would encourage not disputing the answer and awarding the bounty as an appreciation of their effort on your question. However, you are more than welcome to use your right to dispute the answer.

  • M F H M F H
    0

    Thanks. I agree with both: the time I put in is certainly worth more than $30, at the same time I agree that the first approach was somewhat unsatisfying. I believe I have largely improved that, so far concerning behaviour on the first half of the domain, and in particular formulas for the value of m (critical point / maximum). I have some more material for the other half I will try to bring in this evening. (It is 6pm here, maybe already midnight in Europe...)

  • Thank you. It seems that you have proven that h is positive up to q=0.6. However, q also goes from 0.6 to 1, and your solution doesn't say much about that region of the curve. It's hard to quantify "how solved" a partial answer, but simply in terms of q, I guess we could say approximately 60%. Using the 5 steps, we also get 3/5 solved, so that's also around 60%. I hope the last 40% is not that much more difficult! I plan to reinvest whatever comes out back into the bounty.

  • Frankly, I was hoping that a Fourier or Laplace transform or something would pop out a clear answer that $h(q)>0$ for all $0

  • I have submitted the dispute, and I think saying 50-60% of the problem is solved is fair, given that I requested 0

    • M F H M F H
      0

      OK, I understand you dissatisfaction, but: (a) it is NOT true that you asked for an analytical solution, and (b) the current formulation of the question largely benefits from my input : I suggested to introduce the variable q, which is the notation currently used in the area of theta functions, I made that link with the theta function, etc...

    • M F H M F H
      0

      An integral transform could IMHO be used to compute the limits of the two functions separately (viz, lim f(q) = lim g(q) = -1/2 at q=1), but I doubt it can show the positivity. Also note that the question was there for quite a long time (more than 2 weeks) without receiving further comments or other input, when I accepted to answer.

  • the q substitution, theta function, and limit for q=0 were known to me before you got involved: https://www.reddit.com/r/askmath/comments/15ir1sb/prove_this_and_get_your_name_in_a_scientific/ To be clear, I am not in a rush, so if after 2 weeks it expires I'll repost with a higher bounty. This is for my personal curiosity, and ideally to publish a small paper! So I'm not in a rush. If you can help solve the equation maybe we can co-author it!

    • M F H M F H
      0

      Well, I can't see the exact date of the Reddit comment that made that suggestion, but fact is that your question was posed here with a cumbersome e^(-\hat t....) , It remained that way for over 2 weeks, the q and theta did appear exactly when I suggested it. On that Reddit page somebody pretends to give some answers but in the latex file none of the claims is proved. That only shows the difficulty (maybe impossibility?) of an analytic proof. And you didn't disclose any of this information here.

    • M F H M F H
      0

      Okay, I like your final suggestion and I find the problem interesting, so I will continue to work on it. I was just under time pressure right now to deliver something better before 3am today! πŸ™‚

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