1. Introduction and Statement of the Results
- Theorem 1 is a special case of Theorem 5 in Section 3, a more general result that applies to a weight of 1/2 harmonic Maass forms, whose holomorphic parts have algebraic coefficients and whose non-holomorphic parts are period integrals of a weight of unary theta series. The next section will set up all the notation and preliminary results to state and prove the general theorem, as well as how Theorem 1 follows from it.
- The other computable parameters will be described toward the end of Section 3. Briefly, they involve computing the level of a certain half-integral weight modular form from the work of Treneer  as well as the order of vanishing at the cusps; the constants from the results of Lichtenstein ; and, if we do not assume the Generalized Riemann Hypothesis, the constant of Lagarias, Montgomery and Odlyzko .
2. Nuts and Bolts
2.1. Harmonic Maass Forms
- For every , we have:
- We have .
- There is a polynomial, , such that: as for some . Analogous conditions are required at all cusps.
2.2. Elements of the Proof
3. Statement of the General Theorem and Its Proof
- Let Q be an odd prime with for . Then, we have:
- Define and , as in Equation (2.2). Then, there exists a cusp form:
- Define: and let and , as given in Theorem 4 above. Then, the smallest prime, p, for which Equation (3.1) holds satisfies:
- To prove (), apply Theorem 3 to the weakly holomorphic modular form, .
- Statement () is immediate from Theorem 4. ☐
Conflicts of Interest
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