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Hi There, It would help if you gave me some examples of actual positive ternary forms with specific linear dependencies. The main source of dependencies is the Siegel representation formula, which calculates the weighted average of representation numbers in terms of a product of local densities.'' In practice, what this means is that by restricting the target number n to some appropriate arithmetic progression and relating representations by two genera one may get an explicit linear dependence among representation counts. Very much in this spirit is the viewpoint of Jones, given for example in a 1999 paper by Ono and Soundararajan called "Integers Represented by Ternary Quadratic Forms" where they point out that the number of essentially distinct representations of an (eligible) number $N$ by $x^2 + y^2 + 10 z^2$ is just $h( -40 N) / 4$ when $N$ and 10 are coprime. However, the main thing that would surprise me is linear dependence for all n among primitive forms. For instance, Schiemann showed that no two positive ternary forms (inequivalent) have the same theta series.

I'm not sure this facility allows chats back and forth, if you want to try emailing me in person get my address from http://www.ams.org/cml, and for that matter google me as "Will Jagy" in double quotes.

William C. Jagy

JULY: follow-up email sent. I posted something here, JSE felt it might be too revealing. I did not think so, but there is little harm in deleting it and sending you email instead.

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Hi There, It would help if you gave me some examples of actual positive ternary forms with specific linear dependencies. The main source of dependencies is the Siegel representation formula, which calculates the weighted average of representation numbers in terms of a product of local densities.'' In practice, what this means is that by restricting the target number n to some appropriate arithmetic progression and relating representations by two genera one may get an explicit linear dependence among representation counts. Very much in this spirit is the viewpoint of Jones, given for example in a 1999 paper by Ono and Soundararajan called "Integers Represented by Ternary Quadratic Forms" where they point out that the number of essentially distinct representations of an (eligible) number $N$ by $x^2 + y^2 + 10 z^2$ is just $h( -40 N) / 4$ when $N$ and 10 are coprime. However, the main thing that would surprise me is linear dependence for all n among primitive forms. For instance, Schiemann showed that no two positive ternary forms (inequivalent) have the same theta series.

I'm not sure this facility allows chats back and forth, if you want to try emailing me in person get my address from http://www.ams.org/cml, and for that matter google me as "Will Jagy" in double quotes.

William C. Jagy
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