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A natural variant of Lehmer's conjecture that the Ramanujan τ -function never vanishes asks whether, for any given integer α, there exist any n ∈ Z + such that τ (n) = α. A series of recent papers excludes many integers as possible values of the τ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for τ (n). We synthesize these results and methods to prove that if 0 < |α| < 100 and α / ∈ T := {2 k , −24, −48, −70, −90, 92, −96}, then τ (n) = α for all n > 1. Moreover, if α ∈ T and τ (n) = α, then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that |τ (n)| > 100 for all n > 2.

Merca [‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca’s work, we complete the proof of these conjectures.

A classical observation of Deligne shows that, for any prime p ≥ 5, the divisor polynomial of the Eisenstein series E p−1 (z) mod p is closely related to the supersingular polynomial at p,Deuring, Hasse, and Kaneko and Zagier found other families of modular forms which also give the supersingular polynomial at p. In a new approach, we prove an analogue of Deligne's result for the Hecke trace forms T k (z) defined by the Hecke action on the space of cusp forms S k . We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod p, and then relate their divisor polynomials to S p (x) using Deligne's observation.

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