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RITIKA SHARMA
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Volume 129 Issue 1 February 2019 Article ID 0003 Research Article
In this article, we derive an asymptotic formula for the sums of the form $\sum_{n_{1},n_{2}\leq N} f (n_{1}, n_{2})$ with an explicit error term, for any arithmetical function $f$ of two variables with absolutely convergent Ramanujan expansion and Ramanujan coefficients satisfying certain hypothesis.
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Volume 133 All articles Published: 12 January 2023 Article ID 0001 Article
A higher order Levin--Fa\u{i}nle\u{i}b theorem
OLIVIER RAMARE ALISA SEDUNOVA RITIKA SHARMA
When restricted to some non-negative multiplicative function, say $f$, boundedon primes and that vanishes on non square-free integers, our result provides us with an asymptotic for $\sum_{n\leq X} f (n)/n$ with error term $\mathcal{O}(({\rm log} X)^{\kappa - h - 1 + \epsilon}$ (for any positive $\epsilon > 0$) as soon as we have $\sum_{p\leq Q} f (p)(\log p)/p = \kappa \log Q +\eta +O(1/(\log 2Q)^h)$ for a non-negative $\kappa$ and some non-negative integer $h$. The method generalizes the 1967-approach of Levin and Fa\u{i}nle\u{i}b and uses a differential equation.
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Proceedings – Mathematical Sciences
Volume 133, 2023
All articles
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