Riemann’s zeta-function and the divisor problem

Jutila, Matti

doi:10.1007/bf02384301

Cited by 42 publications

(45 citation statements)

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“…[13]. We believe that it could be replaced by 1/4 in view of the analogy between E(t) and the Dirichlet divisor problem (Jutila [14,15]). But we have not been able to prove this.…”

Section: If the Exponent Pair Hypothesis Is True Thenmentioning

confidence: 99%

On higher-power moments of Δ (x)

Zhai

2004

Acta Arith.

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“…[13]. We believe that it could be replaced by 1/4 in view of the analogy between E(t) and the Dirichlet divisor problem (Jutila [14,15]). But we have not been able to prove this.…”

Section: If the Exponent Pair Hypothesis Is True Thenmentioning

confidence: 99%

On higher-power moments of Δ (x)

Zhai

2004

Acta Arith.

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“…The function E * (t) gives an insight into the analogy between the Dirichlet divisor problem and the mean square of |ζ( 1 2 + it)|. It was investigated by several authors, including M. Jutila [15], who introduced the function E * (t), and the author [6]- [8]. Among other things, the author (op.…”

Section: Introductionmentioning

confidence: 99%

“…The technical details are as before, while the function f (t, n) in the sum 1 (T ) is neutralized by using a procedure due to M. Jutila [15, Part II], which was also used in [7]. In this way (5.7) will eventually follow.…”

mentioning

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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“…For the proof of the theorem we will apply the method and some results from [3], see also [7]. Let /~,r (t) = E,~(t2),…”

Section: Proof Follows Immediately From Lemma 1 Using the Definition mentioning

confidence: 99%

A uniform estimate of the error term in the mean square of the Riemann zeta-function

Laurinčikas¹

1995

Lith Math J

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Let ((s), s = cr + it, denote the Riemann zeta-function, and let, for T ~> 2,The asymptotic formulas for I~ (T) as T --+ oo take an important place in the theory of the Riemann zeta-function. A large number of papers is devoted to the case cr = 1/2. Letwhere y denotes the Euler constant. It is known that E(T) = O(T ~)with ot ~< 1/3, and on the other hand it is proved [1] (see also [3]) that E(T) = ~'2(T1/4). The best known result in this direction asserts [2] that E(T) = O (T 7/22+s)for every s > 0.The case (r > 1/2 has been considered in [13]. LetIt has been obtained in [13] that, for 1/2 < cr < 3/4

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Riemann’s zeta-function and the divisor problem

Cited by 42 publications

References 10 publications

On higher-power moments of Δ (x)

On higher-power moments of Δ (x)

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

A uniform estimate of the error term in the mean square of the Riemann zeta-function

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