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Benford’s Law for coefficients of newforms

    https://doi.org/10.1142/S1793042116500299Cited by:3 (Source: Crossref)

    Let f(z)=n=1λf(n)e2πinzSnewk(Γ0(N))f(z)=n=1λf(n)e2πinzSnewk(Γ0(N)) be a newform of even weight k2k2 on Γ0(N)Γ0(N) without complex multiplication. Let denote the set of all primes. We prove that the sequence {λf(p)}p does not satisfy Benford’s Law in any integer base b2. However, given a base b2 and a string of digits S in base b, the set

    Aλf(b,S):={p prime : the first digits of λf(p) in base b are given by S}
    has logarithmic density equal to logb(1+S1). Thus, {λf(p)}p follows Benford’s Law with respect to logarithmic density. Both results rely on the now-proven Sato–Tate Conjecture.

    AMSC: 11F30, 11K06, 11B83
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