Cite as: doi:10.26081/K6K32X

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In the spirit of Lehmer's unresolved speculation on the non-vanishing of
Ramanujan's tau-function, it is natural to ask whether a fixed integer is a
value of τ(n) , or is a Fourier coefficient of any given modular form. In joint
work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained
the first results for such questions. For example, infinitely many spaces are
presented for which the primes l≥37 are not absolute values of coefficients of
any newforms with integer coefficients. For Ramanujan's tau-function, we show
that τ(n) is not in {±l, l < 100 odd prime}.
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