YoungJu Choie

Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>|</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mi>d</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <

Here, a classical observation of Siegel is generalized by determining all the linear relations among the initial Fourier coefficients of a modular form on SL2(Z). As a consequence, spaces Mk are identified, in which there are universal p-divisibility properties for certain p-power coefficients. As a corollary, let f⁡(z)=∑n=1∞af⁡(n)⁢qn∈Sk∩OL⁡[[q]] be a normalized Hecke eigenform (note that q:=e2πiz), and let k ≡ δ(k) (mod 12), where δ(k) ∈ {4, 6, 8, 10, 14}. Reproducing earlier results of Hatada

Let $f=\sum_{n&gt;o} a(n)q^{n}, a(n)$ real, be an arbitrary nonzero cusp form of even integral weight $k\geq 2$ on $\Gamma_0(N),$ square free $N.$ We obtain an effective bound that the first sign change of $a(n)$ occurs.

In this correspondence, we construct theta series that are Jacobi forms, from the complete enumerator of Type II codes over Z/sub 2m/. Moreover, we construct a map from a certain invariant space to the space of Jacobi forms on the full modular group.