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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, which applies to newforms with integer coefficients and trivial residual mod 2 Galois representation, that answers this question for odd integers. We determine infinitely many spaces for which the primes $3leq ellleq 37$ are not absolute values of coefficients of newforms with integer coefficients. For $tau(n)$ with $n>1$, we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 13, pm 17, -19, pm 23, pm 37, pm 691},$$ and assuming GRH we show for primes $ell$ that $$tau(n) ot in left { pm ell : 41leq ellleq 97 {textrm{with}} left(frac{ell}{5}right)=-1right} cup left { -11, -29, -31, -41, -59, -61, -71, -79, -89right}. $$ We also obtain sharp lower bounds for the number of prime factors of such newform coefficients. In the weight aspect, for powers of odd primes $ell$, we prove that $pm ell^m$ is not a coefficient of any such newform $f$ with weight $2k>M^{pm}(ell,m)=O_{ell}(m)$ and even level coprime to $ell,$ where $M^{pm}(ell,m)$ is effectively computable.
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 w
We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a variant of
In this note, we consider the newforms of integral weight, level 4 and of trivial character, and prove that all of them are actually level 1 forms of some non-Dirichlet character. As a byproduct, we can prove that all of them are eigenfunctions of the Fricke involution with eigenvalue -1.
We discuss two different systems of number representations that both can be called base 3/2. We explain how they are connected. Unlike classical fractional extension, these two systems provide a finite representation for integers. We also discuss a c
SIDH is a post-quantum key exchange algorithm based on the presumed difficulty of finding isogenies between supersingular elliptic curves. However, SIDH and related cryptosystems also reveal additional information: the restriction of a secret isogeny