## ON SOME RECENT RESULTS IN THE THEORY OF THE ZETA-FUNCTION (2003)

### BibTeX

@MISC{Ivić03onsome,

author = {Aleksandar Ivić},

title = {ON SOME RECENT RESULTS IN THE THEORY OF THE ZETA-FUNCTION},

year = {2003}

}

### OpenURL

### Abstract

This review article is devoted to the Riemann zeta-function ζ(s), defined for ℜ s> 1 as (1.1) ζ(s) =

### Citations

215 |
The Theory of the Riemann Zeta-Function
- Titchmarsh
- 1951
(Show Context)
Citation Context ...uation to the whole complex plane, its only singularity being the simple pole s = 1 with residue 1. For general information on ζ(s) the reader is referred to the monographs [13], [16], [21], [28] and =-=[51]-=-. From the functional equation (1.2) ζ(s) = χ(s)ζ(1 − s), χ(s) = 2 s π s−1 sin (πs ) Γ(1 − s), 2 which is valid for any complex s, it follows that ζ(s) has zeros at s = −2, −4, . . . . These zeros are... |

140 |
The Riemann Zeta-Function
- Ivić
- 2003
(Show Context)
Citation Context ...ts meromorphic continuation to the whole complex plane, its only singularity being the simple pole s = 1 with residue 1. For general information on ζ(s) the reader is referred to the monographs [13], =-=[16]-=-, [21], [28] and [51]. From the functional equation (1.2) ζ(s) = χ(s)ζ(1 − s), χ(s) = 2 s π s−1 sin (πs ) Γ(1 − s), 2 which is valid for any complex s, it follows that ζ(s) has zeros at s = −2, −4, . ... |

135 |
die Anzahl der Primzahlen unter einer gegbren Grösse, Monatsberichte der Berliner Akadamie
- Riemann, Uber
- 1859
(Show Context)
Citation Context ... 1 arg ζ(1 + iT) = O(log T). π 2 This is the so-called Riemann–von Mangoldt formula. The Riemann hypothesis (henceforth RH for short) is the conjecture, stated by B. Riemann in his epochmaking memoir =-=[49]-=-, that very likely – sehr wahrscheinlich – all complex zeros of ζ(s) have real parts equal to 1 1 2 . For this reason the line σ = 2 is called the “critical line” in the theory of ζ(s). The RH is undo... |

85 | On the distribution of spacing between zeros of the zeta function
- Odlyzko
- 1987
(Show Context)
Citation Context ...d nor disproved. The smallest zeros of ζ(s) (in absolute value) are 1 2 ±14.134725 . . .i. Large scale computations of zeros of ζ(s) have been carried out in recent times by the aid of computers (see =-=[44]-=-–[46], [48]). Suffice to say that it is known today that the first 1.5 billion complex zeros of ζ(s) in the upper half-plane are simple and do have real parts equal to 1 2 , as predicted by the RH. Mo... |

75 |
The difference between consecutive primes
- Baker, Harman, et al.
(Show Context)
Citation Context ...< 1). where pn is the n–th prime (see Ch. 12 of [16]). By combining analytic and sieve techniques various authors have reduced the exponent “7/12”, and the current record holders are Baker and Harman =-=[2]-=-, who proved that pn+1 − pn ≪ p 0.535 n . It is interesting that the so-called “density hypothesis” (3.5) N(σ, T) ≪ T 2−2σ+ε ( 1 2 ≤ σ ≤ 1) gives the bound pn+1 − pn ≪ p 1 2 +ε n , while the much stro... |

52 |
Mean-value theorems in the theory of the Riemann zeta-function
- Ingham
- 1926
(Show Context)
Citation Context ...h moment of the Riemann zeta-function ζ(s) on the critical line is customarily written as (5.1) ∫ T 0 |ζ( 1 2 + it)|4 dt = TP4(log T) + E2(T), P4(x) = 4∑ ajx j . j=0 A classical result of A.E. Ingham =-=[9]-=- (see Ch. 5 of [16] for a relatively simple proof) is that a4 = 1/(2π 2 ) and that the error term E2(T) in (5.1) satisfies the bound E2(T) ≪ T log 3 T. In 1979 D.R. Heath-Brown [4] made significant pr... |

50 |
Spectral Theory of the Riemann Zeta-function. Cambridge
- Motohashi
- 1997
(Show Context)
Citation Context ...E2(T) = O(T 2/3 log C1 T), E2(T) = Ω(T 1/2 ), (5.3) ∫ T E2(t) dt = O(T 3/2 ), 0 ∫ T 0 E 2 2 (t) dt = O(T 2 log C2 T), with effective constants C1, C2 > 0 (the values C1 = 8, C2 = 22 are worked out in =-=[43]-=-). The above results were proved by Y. Motohashi and the author: (5.2) and the first bound in (5.3) in [21],[36] and the second upper bound in (5.3) in [35]. The Ω–result in (5.2) was improved to E2(T... |

44 |
On the difference between consecutive primes
- Huxley
- 1972
(Show Context)
Citation Context ...mprehensive account). In the range 1 3 < σ ≤ the best 2 4 bound is due to A.E. Ingham [10]. This is (3.2) N(σ, T) ≪ T 3(1−σ)/(2−σ) log 5 T, while in the range 3 4 ≤ σ ≤ 1 one has M.N. Huxley’s result =-=[6]-=- that (3.3) N(σ, T) ≪ T 3(1−σ)/(3σ−1) log 44 T. 3When combined, (3.2) and (3.3) yield (3.4) N(σ, T) ≪ T 12(1−σ)/5 log 44 T ( 1 2 From (3.4) one can deduce the bound pn+1 − pn ≪ p 7/12 n log C pn, ≤ σ... |

40 | The distribution and moments of the error term in the Dirichlet divisor problem
- Heath-Brown
- 1992
(Show Context)
Citation Context ...t was proved that ∫ T 0 |E(t)| A dt ≪ T (A+4)/4+ε (0 ≤ A ≤ 35 4 ), 5which in view of the Ω–results is (up to “ε”) optimal, and supports the conjecture E(T) ≪ T 1/4+ε . Later in 1992 D.R. Heath-Brown =-=[5]-=- used these bounds to investigate the distribution function connected with E(T). In the special case A = 2 it is known that (4.2) ∫ T 0 E 2 (t) dt = 2C 3 (2π)−1/2 T 3/2 + O(T log 4 T), C = ∞∑ n=1 d 2 ... |

34 | lattice points and exponential sums
- Huxley, Area
- 1996
(Show Context)
Citation Context ...it was proved by the author [16], [17] that one has unconditionally γn+1 − γn ≪ γ θ+ε n , θ = κ + λ 4(κ + λ) + 2 , where (κ, λ) is a so-called “exponent pair” from the theory of exponential sums (see =-=[8]-=-, [16] for definition), and ε denotes arbitrarily small positive constants. With known results on exponent pairs one obtains then as the strongest known result θ ≤ 0.155945 . . . . Another uncondition... |

30 |
The mean values of the Riemann zeta-function’, LNs 82, Tata Inst. of Fundamental Research, Bombay (distr. by
- Ivić
- 1991
(Show Context)
Citation Context ...omorphic continuation to the whole complex plane, its only singularity being the simple pole s = 1 with residue 1. For general information on ζ(s) the reader is referred to the monographs [13], [16], =-=[21]-=-, [28] and [51]. From the functional equation (1.2) ζ(s) = χ(s)ζ(1 − s), χ(s) = 2 s π s−1 sin (πs ) Γ(1 − s), 2 which is valid for any complex s, it follows that ζ(s) has zeros at s = −2, −4, . . . . ... |

25 |
The mean value of the Riemann zeta-function
- Atkinson
- 1949
(Show Context)
Citation Context ... ∫ T 0 |ζ( 1 2 + it)|2 ( ) T dt = T log + (2γ − 1)T + E(T), 2π where γ = 0.577 . . . is Euler’s constant, and E(T) is to be considered as the error term in the asymptotic formula (4.1). F.V. Atkinson =-=[1]-=- established in 1949 an explicit, albeit complicated formula for E(T), containing two exponential sums of length ≍ T, plus an error term which is O(log 2 T). Atkinson’s formula has been the starting p... |

20 |
A relation between the Riemann zeta-function and the hyperbolic Laplacian
- Motohashi
- 1995
(Show Context)
Citation Context ...d by Y. Motohashi and the author: (5.2) and the first bound in (5.3) in [21],[36] and the second upper bound in (5.3) in [35]. The Ω–result in (5.2) was improved to E2(T) = Ω±(T 1/2 ) by Y. Motohashi =-=[42]-=-. Recently the author [23] made further progress in this problem by proving the following quantitative omega-result: there exist two constants A > 0, B > 1 such that for T ≥ T0 > 0 every interval [T, ... |

19 |
Fractional moments of the Riemann zeta-function, ii
- Heath-Brown
- 1993
(Show Context)
Citation Context ...sult of A.E. Ingham [9] (see Ch. 5 of [16] for a relatively simple proof) is that a4 = 1/(2π 2 ) and that the error term E2(T) in (5.1) satisfies the bound E2(T) ≪ T log 3 T. In 1979 D.R. Heath-Brown =-=[4]-=- made significant progress in this problem by proving that E2(T) ≪ T 7/8+ε . He also calculated a3 = 2(4γ − 1 − log(2π) − 12ζ ′ (2)π −2 )π −2 and produced more complicated expressions for a0, a1 and a... |

17 | On the fourth moment of the Riemann zeta-function, Publs - Ivić - 1995 |

17 |
On the error term for the fourth moment of the Riemann zeta-function
- Ivić
(Show Context)
Citation Context ...y interval [T, BT] contains points T1, T2 for which (5.4) E2(T1) > AT 1/2 1 , E2(T2) < −AT 1/2 2 . Very likely this integral is ∼ CT 2 for some C > 0 as T → ∞. The latest result, proved by the author =-=[32]-=-, complements the upper bound in (5.3) for the mean square, and says that ∫ T 0 E 2 2 (t) dt ≫ T 2 . Very likely this integral is ∼ CT 2 for some C > 0 as T → ∞. In concluding, let it be mentioned tha... |

16 |
On the mean-square of the Riemann zeta-function on the critical line
- Hafner, Ivíc
- 1989
(Show Context)
Citation Context ...y intricate estimation of a certain exponential sum, is due to M.N. Huxley [7]. This is E(T) ≪ T 72/227 (log T) 679/227 , 72 227 = 0.3171806 . . . . In the other direction, J.L. Hafner and the author =-=[3]-=-, [21] proved in 1987 that there exist absolute constants A, B > 0 such that E(T) = Ω+ { (T log T) 1/4 (log log T) (3+log 4)/4 e −A√ loglog log T } and E(T) = Ω− { T 1/4 ( 1/4 B(log log T) exp (log lo... |

15 |
The fourth moment of the Riemann zeta-function
- Ivić, Motohashi
(Show Context)
Citation Context ...2 T), with effective constants C1, C2 > 0 (the values C1 = 8, C2 = 22 are worked out in [43]). The above results were proved by Y. Motohashi and the author: (5.2) and the first bound in (5.3) in [21],=-=[36]-=- and the second upper bound in (5.3) in [35]. The Ω–result in (5.2) was improved to E2(T) = Ω±(T 1/2 ) by Y. Motohashi [42]. Recently the author [23] made further progress in this problem by proving t... |

15 |
The mean square of the error term for the fourth moment of the zeta-function
- Ivić, Motohashi
(Show Context)
Citation Context ...he values C1 = 8, C2 = 22 are worked out in [43]). The above results were proved by Y. Motohashi and the author: (5.2) and the first bound in (5.3) in [21],[36] and the second upper bound in (5.3) in =-=[35]-=-. The Ω–result in (5.2) was improved to E2(T) = Ω±(T 1/2 ) by Y. Motohashi [42]. Recently the author [23] made further progress in this problem by proving the following quantitative omega-result: ther... |

12 |
The Mellin transform and the Riemann zeta-function
- Ivić
(Show Context)
Citation Context ... f = Ω+ means that lim sup f/g > 0, f = Ω− means that lim inf f/g < 0, and f = Ω±(g) means that lim supf/g > 0 and lim inf f/g < 0 both hold. A quantitative Ω–result for E(T) was proved by the author =-=[23]-=-: There exist constants C, D > 0 such that for T ≥ T0 every interval [T, T + CT 1/2 ] contains points t1, t2 for which E(t1) > Dt 1/4 1 , E(t2) < −Dt 1/4 2 . Numerical calculations pertaining to E(T) ... |

11 |
On the ternary additive divisor problem and the sixth moment of the zeta-function, Sieve Methods, Exponential Sums, and their Applications
- Ivić
- 1997
(Show Context)
Citation Context ...for the mean square, and says that ∫ T 0 E 2 2 (t) dt ≫ T 2 . Very likely this integral is ∼ CT 2 for some C > 0 as T → ∞. In concluding, let it be mentioned that the sixth moment was investigated in =-=[29]-=-, where it was shown that ∫ T 0 |ζ( 1 2 + it)|6 dt ≪ε T 1+ε does hold if a certain conjecture involving the so-called ternary additive divisor problem is true. On the other hand, it is known that (see... |

8 |
On the estimation of N(σ
- Ingham
- 1940
(Show Context)
Citation Context ...uently improved. In obtaining upper bounds for N(σ, T) one can use several techniques (see Ch. 11 of [16] for a comprehensive account). In the range 1 3 < σ ≤ the best 2 4 bound is due to A.E. Ingham =-=[10]-=-. This is (3.2) N(σ, T) ≪ T 3(1−σ)/(2−σ) log 5 T, while in the range 3 4 ≤ σ ≤ 1 one has M.N. Huxley’s result [6] that (3.3) N(σ, T) ≪ T 3(1−σ)/(3σ−1) log 44 T. 3When combined, (3.2) and (3.3) yield ... |

8 |
Large values of the error term in the divisor problem
- Ivić
- 1983
(Show Context)
Citation Context ...t1) > Dt 1/4 1 , E(t2) < −Dt 1/4 2 . Numerical calculations pertaining to E(T) have been carried out in [37] by the author and H. te Riele. Power moments of E(T) were considered in 1983 by the author =-=[12]-=-, where it was proved that ∫ T 0 |E(t)| A dt ≪ T (A+4)/4+ε (0 ≤ A ≤ 35 4 ), 5which in view of the Ω–results is (up to “ε”) optimal, and supports the conjecture E(T) ≪ T 1/4+ε . Later in 1992 D.R. Hea... |

8 | Recent developments in the mean square theory of the Riemann zeta and other zeta-functions
- Matsumoto
- 2000
(Show Context)
Citation Context ...ean square formulas are concerned, one can distinguish between the cases σ = 1 2 and 1 2 < σ ≤ 1. Here we shall briefly discuss only the former 4case, and for the latter we refer the reader to [34], =-=[41]-=-. One has the asymptotic formula (4.1) ∫ T 0 |ζ( 1 2 + it)|2 ( ) T dt = T log + (2γ − 1)T + E(T), 2π where γ = 0.577 . . . is Euler’s constant, and E(T) is to be considered as the error term in the as... |

7 |
On the zeros of Riemann's zeta-function
- Selberg
- 1942
(Show Context)
Citation Context ...ros on the critical line. Let N0(T) denote the number of zeros of ζ(s) of the form 1 2 + it, 0 < t ≤ T. The RH is in fact the statement that N0(T) = N(T) for T > 0. A fundamental result of A. Selberg =-=[50]-=- from 1942 states that (2.1) N0(T) > CN(T) for T ≥ T0 and some positive constant C > 0. In other words, a positive proportion of all complex zeros of ζ(s) lies on the critical line. A substantial adva... |

6 |
On some results concerning the Riemann hypothesis, in: Y
- Ivic
- 1998
(Show Context)
Citation Context ...ions (Dirichlet series) sharing similar properties with ζ(s). Despite much evidence in favour of the RH, there are also some reasons to be skeptical about its truth – see, for example, [24], [27] and =-=[30]-=-. The aim of this paper is to present briefly some recent results in the theory of ζ(s). The choice of subjects is motivated by the limited length of this text and by the author’s personal research in... |

6 | Analytic computations in number theory - Odlyzko - 1994 |

5 |
Exponential sums with a difference
- Heath-Brown, Huxley
- 1990
(Show Context)
Citation Context ...t E(T) ≪ T 1/4+ε , and this bound cannot be proved even if the RH is assumed. The best known upper bound for E(T), obtained by intricate estimation of a certain exponential sum, is due to M.N. Huxley =-=[7]-=-. This is E(T) ≪ T 72/227 (log T) 679/227 , 72 227 = 0.3171806 . . . . In the other direction, J.L. Hafner and the author [3], [21] proved in 1987 that there exist absolute constants A, B > 0 such tha... |

4 |
Gaps between consecutive zeros of the Riemann zeta-function on the critical line, Monatsh
- Ivic, Jutila
- 1988
(Show Context)
Citation Context ... − γn ≪ 2 1 log log γnfor the gap between consecutive zeros on the critical line. The bound (2.3) is certainly out of reach at present. For some unconditional results on γn+1 − γn, see [17]–[19] and =-=[33]-=-. For example, it was proved by the author [16], [17] that one has unconditionally γn+1 − γn ≪ γ θ+ε n , θ = κ + λ 4(κ + λ) + 2 , where (κ, λ) is a so-called “exponent pair” from the theory of exponen... |

3 |
On the error term in the mean square formula for the Riemann zeta-function in the critical strip
- Ivić, Matsumoto
(Show Context)
Citation Context ...r as mean square formulas are concerned, one can distinguish between the cases σ = 1 2 and 1 2 < σ ≤ 1. Here we shall briefly discuss only the former 4case, and for the latter we refer the reader to =-=[34]-=-, [41]. One has the asymptotic formula (4.1) ∫ T 0 |ζ( 1 2 + it)|2 ( ) T dt = T log + (2γ − 1)T + E(T), 2π where γ = 0.577 . . . is Euler’s constant, and E(T) is to be considered as the error term in ... |

2 | On a problem connected with zeros of ζ(s) on the critical line, Monatshefte Math - Ivić - 1987 |

2 | On sums of gaps between the zeros of ζ(s) on the critical - Ivić |

2 |
Zero-density estimates for L−functions, Acta Arith., 32
- Jutila
- 1977
(Show Context)
Citation Context ... 2 n log pn, which is still insufficient to prove the old conjecture: between every two squares there is always a prime. The bound (3.5) is known to hold for σ ≥ 11 14 , which was proved by M. Jutila =-=[38]-=- in 1977. For various values of σ in the range 3 < σ < 1 sharper results than 4 (3.3) or (3.4) have been proved. For example, the author [11], [14]–[16] obtained the bounds N(σ, T) ≪ T 3(1−σ)/(2σ)+ε N... |

2 |
la moyenne de la fonction zêta
- Sur
- 1993
(Show Context)
Citation Context ... E 2 (t) dt = 2C 3 (2π)−1/2 T 3/2 + O(T log 4 T), C = ∞∑ n=1 d 2 (n)n −3/2 where d(n) is the number of divisors of n. The bound for the error term in (4.2) was obtained independently by E. Preissmann =-=[47]-=- and the author [21]. In 1992 K.-M. Tsang [52] proved that, for some δ > 0 and explicit constants c1, c2 > 0, ∫ T 0 ∫ T 0 E 3 (t) dt = c1T 7 4 + O(T 7 4 −δ ), E 4 (t) dt = c2T 2 + O(T 2−δ ). The autho... |

2 |
de Lune, Computational number theory at CWI
- Riele, van
- 1970
(Show Context)
Citation Context ...oved. The smallest zeros of ζ(s) (in absolute value) are 1 2 ±14.134725 . . .i. Large scale computations of zeros of ζ(s) have been carried out in recent times by the aid of computers (see [44]–[46], =-=[48]-=-). Suffice to say that it is known today that the first 1.5 billion complex zeros of ζ(s) in the upper half-plane are simple and do have real parts equal to 1 2 , as predicted by the RH. Moreover, man... |

1 |
A note on the zero-density estimates for the zeta function
- Ivić
(Show Context)
Citation Context ...nown to hold for σ ≥ 11 14 , which was proved by M. Jutila [38] in 1977. For various values of σ in the range 3 < σ < 1 sharper results than 4 (3.3) or (3.4) have been proved. For example, the author =-=[11]-=-, [14]–[16] obtained the bounds N(σ, T) ≪ T 3(1−σ)/(2σ)+ε N(σ, T) ≪ T 9(1−σ)/(7σ−1)+ε N(σ, T) ≪ T 6(1−σ)/(5σ−1)+ε ( 3831 4791 ( 41 53 ( 13 17 = 0.799624 . . . ≤ σ ≤ 1), = 0.773585 . . . ≤ σ ≤ 1), = 0.... |

1 |
Topics in recent zeta-function theory
- Ivić
- 1983
(Show Context)
Citation Context ...t admits meromorphic continuation to the whole complex plane, its only singularity being the simple pole s = 1 with residue 1. For general information on ζ(s) the reader is referred to the monographs =-=[13]-=-, [16], [21], [28] and [51]. From the functional equation (1.2) ζ(s) = χ(s)ζ(1 − s), χ(s) = 2 s π s−1 sin (πs ) Γ(1 − s), 2 which is valid for any complex s, it follows that ζ(s) has zeros at s = −2, ... |

1 |
Exponent pairs and power moments of the zeta-function
- Ivić
- 1981
(Show Context)
Citation Context ...o hold for σ ≥ 11 14 , which was proved by M. Jutila [38] in 1977. For various values of σ in the range 3 < σ < 1 sharper results than 4 (3.3) or (3.4) have been proved. For example, the author [11], =-=[14]-=-–[16] obtained the bounds N(σ, T) ≪ T 3(1−σ)/(2σ)+ε N(σ, T) ≪ T 9(1−σ)/(7σ−1)+ε N(σ, T) ≪ T 6(1−σ)/(5σ−1)+ε ( 3831 4791 ( 41 53 ( 13 17 = 0.799624 . . . ≤ σ ≤ 1), = 0.773585 . . . ≤ σ ≤ 1), = 0.764705... |

1 | A zero-density theorem for the Riemann zeta-function - Ivić - 1984 |

1 | On consecutive zeros of the Riemann zeta-function on the critical line. Sémin. de Théorie des Nombres, Université de Bordeaux 1986/87, Exposé no - Ivić |

1 |
zéros de la fonction zeta de Riemann sur la droite critique, Groupe de travail en théorie analytique et élementaire des nombres 1986-1987, Université ParisSud
- Ivić, Les
(Show Context)
Citation Context ...2.3) γn+1 − γn ≪ 2 1 log log γnfor the gap between consecutive zeros on the critical line. The bound (2.3) is certainly out of reach at present. For some unconditional results on γn+1 − γn, see [17]–=-=[19]-=- and [33]. For example, it was proved by the author [16], [17] that one has unconditionally γn+1 − γn ≪ γ θ+ε n , θ = κ + λ 4(κ + λ) + 2 , where (κ, λ) is a so-called “exponent pair” from the theory o... |

1 |
On a class of convolution functions connected with ζ(s
- Ivić
(Show Context)
Citation Context ...ther zeta-functions (Dirichlet series) sharing similar properties with ζ(s). Despite much evidence in favour of the RH, there are also some reasons to be skeptical about its truth – see, for example, =-=[24]-=-, [27] and [30]. The aim of this paper is to present briefly some recent results in the theory of ζ(s). The choice of subjects is motivated by the limited length of this text and by the author’s perso... |

1 |
On the distribution of zeros of a class of convolution functions
- Ivić
(Show Context)
Citation Context ...eta-functions (Dirichlet series) sharing similar properties with ζ(s). Despite much evidence in favour of the RH, there are also some reasons to be skeptical about its truth – see, for example, [24], =-=[27]-=- and [30]. The aim of this paper is to present briefly some recent results in the theory of ζ(s). The choice of subjects is motivated by the limited length of this text and by the author’s personal re... |

1 |
Uvod u Analitičku Teoriju Brojeva, Izdavačka knjiˇzarnica Zorana
- Ivić
- 1996
(Show Context)
Citation Context ...ic continuation to the whole complex plane, its only singularity being the simple pole s = 1 with residue 1. For general information on ζ(s) the reader is referred to the monographs [13], [16], [21], =-=[28]-=- and [51]. From the functional equation (1.2) ζ(s) = χ(s)ζ(1 − s), χ(s) = 2 s π s−1 sin (πs ) Γ(1 − s), 2 which is valid for any complex s, it follows that ζ(s) has zeros at s = −2, −4, . . . . These ... |

1 |
On some problems involving the mean square of |ζ( 1 +it)|, Bulletin CXVI de l’Académie Serbe des Sciences et des Arts
- Ivić
- 1998
(Show Context)
Citation Context ...nd the author [21]. In 1992 K.-M. Tsang [52] proved that, for some δ > 0 and explicit constants c1, c2 > 0, ∫ T 0 ∫ T 0 E 3 (t) dt = c1T 7 4 + O(T 7 4 −δ ), E 4 (t) dt = c2T 2 + O(T 2−δ ). The author =-=[31]-=- recently improved these bounds to O(T 47 28 +ε ) and O(T 45 23 +ε ), respectively. 5. The mean fourth power The asymptotic formula for the fourth moment of the Riemann zeta-function ζ(s) on the criti... |

1 |
Riele, On the zeros of the error term for the mean square of |ζ( 1
- Ivić, te
(Show Context)
Citation Context ..., D > 0 such that for T ≥ T0 every interval [T, T + CT 1/2 ] contains points t1, t2 for which E(t1) > Dt 1/4 1 , E(t2) < −Dt 1/4 2 . Numerical calculations pertaining to E(T) have been carried out in =-=[37]-=- by the author and H. te Riele. Power moments of E(T) were considered in 1983 by the author [12], where it was proved that ∫ T 0 |E(t)| A dt ≪ T (A+4)/4+ε (0 ≤ A ≤ 35 4 ), 5which in view of the Ω–res... |

1 | de Lune, H.J.J. te Riele and - van - 1987 |

1 |
The 10 20 -th zero of the Riemann zeta-function and 175 million of its neighbors
- Odlyzko
(Show Context)
Citation Context ... disproved. The smallest zeros of ζ(s) (in absolute value) are 1 2 ±14.134725 . . .i. Large scale computations of zeros of ζ(s) have been carried out in recent times by the aid of computers (see [44]–=-=[46]-=-, [48]). Suffice to say that it is known today that the first 1.5 billion complex zeros of ζ(s) in the upper half-plane are simple and do have real parts equal to 1 2 , as predicted by the RH. Moreove... |

1 |
Higher power moments of δ(x), E(t) and P(x
- Tsang
(Show Context)
Citation Context ...T), C = ∞∑ n=1 d 2 (n)n −3/2 where d(n) is the number of divisors of n. The bound for the error term in (4.2) was obtained independently by E. Preissmann [47] and the author [21]. In 1992 K.-M. Tsang =-=[52]-=- proved that, for some δ > 0 and explicit constants c1, c2 > 0, ∫ T 0 ∫ T 0 E 3 (t) dt = c1T 7 4 + O(T 7 4 −δ ), E 4 (t) dt = c2T 2 + O(T 2−δ ). The author [31] recently improved these bounds to O(T 4... |