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In this paper: "ON A RESULT OF G.PÓLYA CONCERNING THE RIEMANN $\xi - FUNCTION$ by DENNIS A. HEJHAL"

the author defines

$$ \theta(x)=\sum_{n=-\infty}^{\infty}e^{-\pi n^{2} x} $$ then he says, in the begining of the second page: "Since $e^{u/2} \theta(e^{2u})$" is even, so is ..."

I Can't seen why $e^{u/2} \theta(e^{2u})$ is even. He is saying that the following function is even

$$e^{u/2}\sum_{n=-\infty}^{\infty}e^{-\pi n^{2}e^{2u}}$$

I used mathematica and got the following result Imege plot

Clearly this is not even. Certainly I'm missing something obvious, or is this a typo? If so what should have been written?

Thanks.

After the remark of Professor Vetor, I produced computations with more terms, and indeed it seems that it is even. Amazing!! enter image description here

Neves
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    If you have a link to the paper that would be useful - I don't see it on the arxiv offhand... – Steven Stadnicki Jan 14 '18 at 21:16
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    That's an artifact: for negative $u$, your series doesn't converge rapidly enough, summing form $-1024$ to $1024$ is not accurate enough. It's a special property of that function based on the Poison summation formula, it can also be used to prove the famous functional equation of the zeta function. –  Jan 14 '18 at 21:19
  • @ProfessorVector, could you produce a proof of this as an answer so that I can accept it? – Neves Jan 14 '18 at 22:01
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    This is a special case of Jacobi identities for theta functions: https://en.wikipedia.org/wiki/Theta_function#Jacobi_identities See also: https://en.wikipedia.org/wiki/Theta_function#Relation_to_the_Riemann_zeta_function – Did Jan 14 '18 at 22:13

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The clue is the use of $\theta$ for the function. This is a variant of a Jacobi theta function. More precisely, $\;\theta(x)=\theta_3(0,e^{-\pi x}).\;$ By the transformation properties of $\theta_3$ we have $\theta(x)=\theta(1/x)/\sqrt{x},\;$ and combined with $\;e^{-u}=1/e^u,\;\sqrt{e^u}=e^{u/2},\;$ the even property follows.

Somos
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