The mean square of the divisor function

• Autorzy: Chaohua Jia , Ayyadurai Sankaranarayanan
• Języki publikacji: EN

Abstrakty

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
$E(x) = O(x^{1/2}(log x)⁵)$.

Kategorie tematyczne

• 11M06: ζ ( s ) and L ( s , χ )

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 164
• Numer: 2
• Strony: 181-208

Daty

wydano

2014

Twórcy

autor

Chaohua Jia

• Institute of Mathematics, Academia Sinica, Beijing 100190, P.R. China
• Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, P.R. China

autor

Ayyadurai Sankaranarayanan

• School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India

Identyfikatory

DOI

10.4064/aa164-2-7