### Jameson, M, Thorner, J and Ye, L (2014)

#### Benford's Law for Coefficients of Newforms

arXiv:1407.1577 [math.NT]; posted July 7, 2014; last accessed November 10, 2014.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**Abstract:** Let f(z)=∑∞n=1λf(n)e2πinz∈Snewk(Γ0(N)) be a normalized Hecke eigenform of even weight k≥2 on Γ0(N) without complex multiplication. Let ℙ denote the set of all primes. We prove that the sequence {λf(p)}p∈ℙ does not satisfy Benford's Law in any base b≥2. However, given a base b≥2 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+S−1). Thus {λf(p)}p∈ℙ follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.

**Bibtex:**

```
@unpublished{,
AUTHOR = {Jameson, Marie and Thorner, Jesse and Ye, Lynnelle},
MONTH = {July},
NOTE = {last accessed November 10, 2014},
TITLE = {Benford's Law for Coefficients of Newforms},
YEAR = {2014},
DATE = {July 7, 2014},
EPRINT = {arXiv:1407.1577 [math.NT], %% e.g., quant-ph/9812037},
URL = {http://arxiv.org/abs/1407.1577}
}
```

**Reference Type:** E-Print

**Subject Area(s):** Analysis, Statistics