2003
DOI: 10.4064/aa109-4-5
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On an asymptotic formula of Srinivasa Ramanujan

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Cited by 25 publications
(22 citation statements)
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“…The error term in (3.3) can be improved to O(x 1/2 log 5 x log log x) (see Ramachandra-Sankaranarayanan [18]), but the exponent 1/2 of x cannot be improved without assumptions on the zero-free region of ζ(s) (such as e.g., the Riemann hypothesis that all complex zeros of ζ(s) have real parts equal to 1/2). However, this improvement is not needed in view of the error term O(HU 1/2 T ε ) in (1.7).…”
Section: The Proof Of Theorem 1 and Theoremmentioning
confidence: 99%
“…The error term in (3.3) can be improved to O(x 1/2 log 5 x log log x) (see Ramachandra-Sankaranarayanan [18]), but the exponent 1/2 of x cannot be improved without assumptions on the zero-free region of ζ(s) (such as e.g., the Riemann hypothesis that all complex zeros of ζ(s) have real parts equal to 1/2). However, this improvement is not needed in view of the error term O(HU 1/2 T ε ) in (1.7).…”
Section: The Proof Of Theorem 1 and Theoremmentioning
confidence: 99%
“…For the case of the Riemann zeta-function, this is contained in Ramachandra [10], Theorem 1. Our argument follows the lines of Lemma 3.2 in Ramachandra and Sankaranarayanan [11].…”
Section: Some Auxiliary Resultsmentioning
confidence: 77%
“…In addition, our analysis has been inspired by recent work (3) of K. Ramachandra and A. Sankaranarayanan [11] which deals with (2) Therefore, in these cases, the assumption of the truth of the Riemann Hypothesis (RH) leads to a better error term O(x θ ) , with some θ < 1 2 . In contrast, our Theorem is as sharp as it would be if RH could be proven.…”
Section: Introductionmentioning
confidence: 99%
“…This part is just a simple supplement of [12]. The main body of the present paper is the proof of Theorems 2 and 3, for which we apply the method of a paper of Ramachandra and the second author [15]. In Section 3 we will give a lemma, which is a generalization of Lemma 3.2 of [15].…”
Section: Introduction and Statement Of Main Resultsmentioning
confidence: 99%