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The mean square of the divisor function

100%
Jia C., Sankaranarayanan A.
Acta Arithmetica
|
2014
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tom 164
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nr 2
181-208
EN
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 $E(x) = O(x^{1/2}(log x)⁵)$.
2
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The distribution of Fourier coefficients of cusp forms over sparse sequences

100%
Lao H., Sankaranarayanan A.
Acta Arithmetica
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2014
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tom 163
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nr 2
101-110
EN
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.
3
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On the Riesz means of n/ϕ(n) - III

100%
Sankaranarayanan A., Singh S.
Acta Arithmetica
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2015
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tom 170
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nr 3
275-286
EN
Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term $E_k(x)$ where $1/k! ∑_{n≤x} n/ϕ(n) (1 - n/x)^{k} = M_k(x) + E_k(x)$. For instance, the upper bound for |E_k(x)| established here improves the earlier known upper bounds for all integers k satisfying $k ≫ (log x)^{1+ϵ}$.
4
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On the moments of the Carmichael λ function

100%
Luca F., Sankaranarayanan A.
Acta Arithmetica
|
2006
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tom 123
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nr 4
389-398
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