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We consider natural variants of Lehmers unresolved conjecture that Ramanujans tau-function never vanishes. Namely, for $n>1$ we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 691}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujans well-known congruences for $tau(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations.
In the spirit of Lehmers unresolved speculation on the nonvanishing of Ramanujans tau-function, it is natural to ask whether a fixed integer is a value of $tau(n)$ or is a Fourier coefficient $a_f(n)$ of any given newform $f(z)$. We offer a method, w
Let $K$ be a local function field of characteristic $l$, $mathbb{F}$ be a finite field over $mathbb{F}_p$ where $l e p$, and $overline{rho}: G_K rightarrow text{GL}_n (mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin metho
In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirlings approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his resu
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we pr
Using Ramanujans identities and the Weierstrass-Enneper representation of minimal surfaces and the analogue for Born-Infeld solitons, we derive further non-trivial identities.