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3 Acta Arithmetica

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The distribution of Fourier coefficients of cusp forms over sparse sequences

100%

Lao H. , Sankaranarayanan A.

Acta Arithmetica

2014

tom 163

nr 2

101-110

Let $λ_f(n)$ be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform $f(z) ∈ S_{k}(Γ)$. We establish that $∑_{n ≤ x}λ_f^2(n^j) = c_{j} x + O(x^{1-2/((j+1)^2+1)})$ for j = 2,3,4, which improves the previous results. For j = 2, we even establish a better result.

The mean square of the divisor function

100%

Jia C. , Sankaranarayanan A.

Acta Arithmetica

2014

tom 164

nr 2

181-208

Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that $∑_{n≤x} d²(n) = xP(log x) + E(x)$, where P(y) is a cubic polynomial in y and $E(x) = O(x^{3/5 + ε})$, with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E(x)=O(x^{1/2 + ε})$. In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E(x) = O(x^{1/2}(log x)⁵loglog x)$. In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E(x) = O(x^{1/2}(log x)⁵)$.

On the moments of the Carmichael λ function

100%

Luca F. , Sankaranarayanan A.

Acta Arithmetica

2006

tom 123

nr 4

389-398