On an asymptotic formula of Srinivasa Ramanujan

Ramachandra, K.; Sankaranarayanan, A.

doi:10.4064/aa109-4-5

Cited by 25 publications

(22 citation statements)

References 10 publications

(9 reference 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%

On the divisor function and the Riemann zeta-function in short intervals

Ivić

2008

Ramanujan J

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Abstract. We obtain, for T ε ≤ U = U(T ) ≤ T 1/2−ε , asymptotic formulas forwhere ∆(x) is the error term in the classical divisor problem, and E(T ) is the error term in the mean square formula for |ζ( 1 2 + it)|. Upper bounds of the form O ε (T 1+ε U 2 ) for the above integrals with biquadrates instead of square are shown to hold for T 3/8 ≤ U = U(T ) ≪ T 1/2 . The connection between the moments of E(t + U) − E(t) and |ζ( 1 2 + it)| is also given. Generalizations to some other number-theoretic error terms are discussed. IntroductionPower moments represent one of the most important parts of the theory of the Riemann zeta-function ζ(s), defined asand otherwise by analytic continuation. Of particular significance are the moments on the "critical line" σ = 1 2 , and a vast literature exists on this subject (see e.g., the monographs [5], [6], and [23]). In this paper we shall be concerned with moments of the error function1991 Mathematics Subject Classification. 11 M 06, 11 N 37. Key words and phrases. The Riemann zeta-function, the divisor functions, power moments in short intervals, upper bounds.Typeset by A M S-T E X

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Section: The Proof Of Theorem 1 and Theoremmentioning

confidence: 99%

On the divisor function and the Riemann zeta-function in short intervals

Ivić

2008

Ramanujan J

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“…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%

The average number of solutions of the Diophantine equation U²+V²=W³and related arithmetic functions

Kühleitner

Nowak

2004

Acta Mathematica Hungarica

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“…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%

On the mean square of standard L-functions attached to Ikeda lifts

Matsumoto

Sankaranarayanan

2006

Math. Z.

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By using estimates on the frequency of large values of the Riemann zetafunction and modular L-functions attached to the full modular group SL(2, Z), we prove sharp upper and lower estimates of the mean square of standard L-functions attached to Siegel cusp forms which are Ikeda lifts, on boundaries and the central line of the critical strip. The mean square of spinor L-functions attached to Saito-Kurokawa lifts is also studied.

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On an asymptotic formula of Srinivasa Ramanujan

Cited by 25 publications

References 10 publications

On the divisor function and the Riemann zeta-function in short intervals

On the divisor function and the Riemann zeta-function in short intervals

The average number of solutions of the Diophantine equation U²+V²=W³and related arithmetic functions

On the mean square of standard L-functions attached to Ikeda lifts

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On an asymptotic formula of Srinivasa Ramanujan

Cited by 25 publications

References 10 publications

On the divisor function and the Riemann zeta-function in short intervals

On the divisor function and the Riemann zeta-function in short intervals

The average number of solutions of the Diophantine equation U2+V2=W3and related arithmetic functions

On the mean square of standard L-functions attached to Ikeda lifts

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Product

Resources

About

The average number of solutions of the Diophantine equation U²+V²=W³and related arithmetic functions