## ASYMPTOTIC ESTIMATION OF ξ (2n) (1/2): ON A CONJECTURE OF FARMER AND RHOADES

### BibTeX

@MISC{Coffey_asymptoticestimation,

author = {Mark W. Coffey},

title = {ASYMPTOTIC ESTIMATION OF ξ (2n) (1/2): ON A CONJECTURE OF FARMER AND RHOADES},

year = {}

}

### OpenURL

### Abstract

Abstract. We verify a very recent conjecture of Farmer and Rhoades on the asymptotic rate of growth of the derivatives of the Riemann xi function at s =1/2. We give two separate proofs of this result, with the more general method not restricted to s =1/2. We briefly describe other approaches to our results, give a heuristic argument, and mention supporting numerical evidence.

### Citations

175 |
Asymptotic Approximations of Integrals
- Wong
- 1989
(Show Context)
Citation Context ...)]dy, where f(j) > 2 is assumed to be monotonically increasing in j. Wealsoimposethe condition: if b<0 (respectively b>0), then a>1 − b (respectively a<1 − b). We apply the method of steepest descent =-=[21]-=-. We write Ij = � ∞ 0 eg(y) dy and ω(x) =e −πx [1 + ω2(x)] = e −πx [1 + O(e −3πx )]. Define y ∗ as the solution of thes4 MARK W. COFFEY equation dg/dy =0. Wehave dg − 2 2(a + b − 1)by + f(a +2b− 1) = ... |

135 |
Multiplicative Number Theory
- Davenport
- 1980
(Show Context)
Citation Context ... and mention supporting numerical evidence. Introduction Let ξ be the Riemann xi function, given by ξ(s) = 1 2s(s − 1)π−s/2Γ(s/2)ζ(s), where Γ is the Gamma function and ζ is the Riemann zeta function =-=[7, 17]-=-. It satisfies the functional equation ξ(s) = ξ(1 − s) and is entire of order 1. The functional equation implies that all odd order derivatives of ξ vanish at s =1/2. On the other hand, estimation of ... |

70 |
Titchmarsh, The Theory of the Riemann Zeta-Function
- C
- 1986
(Show Context)
Citation Context ... and mention supporting numerical evidence. Introduction Let ξ be the Riemann xi function, given by ξ(s) = 1 2s(s − 1)π−s/2Γ(s/2)ζ(s), where Γ is the Gamma function and ζ is the Riemann zeta function =-=[7, 17]-=-. It satisfies the functional equation ξ(s) = ξ(1 − s) and is entire of order 1. The functional equation implies that all odd order derivatives of ξ vanish at s =1/2. On the other hand, estimation of ... |

54 |
A generalisation of Stirling’s formula
- Hayman
- 1956
(Show Context)
Citation Context ...e derivatives of the xi function. In particular, we briefly elaborate on important earlier numerical calculations. Proof of Proposition 1 The result (1) uses estimates of Grosswald [9, 10] and Hayman =-=[13]-=- and the notation here essentially follows that of Grosswald. The function Ξ(t) isevenand the coefficients of t2 alternate in sign. Putting ∞� (3) ξ(1/2+it) ≡ Ξ(t)= cnt n = f(−t 2 ∞� )=f(z)= αnz n , n... |

23 |
Asymptotic expansions for coefficients of analytic functions
- Harris, Shoenfeld
(Show Context)
Citation Context ...admissible” functions, since he further restricts the allowable functions. Very related work on asymptotic expansions for the coefficients of analytic functions was performed by Harris and Schoenfeld =-=[11]-=-, wherein they invokesASYMPTOTIC ESTIMATE OF ξ (2n) (1/2) 5 weaker hypotheses than Grosswald. Pertinent results on the maximum modulus and Fourier transform of the Riemann xi function are given by Hav... |

14 |
Relations and positivity results for derivatives of the Riemann ξ function
- Coffey
- 2004
(Show Context)
Citation Context ...s Corollary 1. (11) ln Dn =4[1−ln(4n)+ln(lnn)]n− 4n 7 3 + ln(2n)− ln(ln n)+O(1), n→∞. ln n 2 2 Again, Proposition 2 offers refinements to this result. Proof of Proposition 2 We have for all complex s =-=[1]-=- (12) ξ (j) (s)= 1 2 j−1 � ∞ [x 1 3/2 ω ′ (x)] ′ [x s/2−1/2 +(−1) j x −s/2 ]ln j xdx, where the theta function ω(x) ≡ �∞ n=1 exp(−πn2x) [7]. We put, for a>1, a+b �=1, and b �=0, (13) (14) = (15) � ∞ =... |

9 |
Scienti computation on mathematical problems and conjectures
- Varga
- 1990
(Show Context)
Citation Context ...hat simply the leading term of (1) suffices to verify the conjecture of Farmer and Rhoades. An initial set of values � ln ξ (2n) (1/2) �20 is effectively given in Table 4.1 of [6] n=0 or Table 3.1 of =-=[18]-=-. However, these values are not yet in the asymptotic regime and indeed show a decrease rather than an increase with n. Proposition 2 is separately proved from Proposition 1 and obviously subsumes the... |

8 | Differentiation evens out zero spacings
- Farmer, Rhoades
(Show Context)
Citation Context ...(−1) (j − 2) j � � � s−1/2 ln(j − 2) j − 2 � � � � � �� �j−3/2 � � j − 2 j − 2 (j − 2) (2) × ln − ln ln + o(1) exp − . π π ln(j − 2) Proposition 1 is in response to a conjecture of Farmer and Rhoades =-=[8]-=- that ln ξ (2n) (1/2) should increase very regularly and not too much faster than linearly as n →∞. They made this conjecture in the course of a study of the effect of repeated differentiation upon th... |

3 |
New results concerning power series expansions of the Riemann xi function and the Li/Keiper constants, preprint
- Coffey
- 2005
(Show Context)
Citation Context ...ms of the well-known Stieltjes constants γk (e.g., [2, 3]). In particular, for (30), the values ξ (j) (1) (cf. Corollary 2) precisely enter the Li criterion for the validity of the Riemann hypothesis =-=[1, 4]-=-. Our Proposition 2 is strong enough to provide an alternative proof of the conjecture (25) confirmed by Ki [14]. It also gives asymptotic information on many other important constants. In particular,... |

3 |
Generalization of a formula of Hayman, and its application to the study of Riemann’s zeta function
- Grosswald
- 1966
(Show Context)
Citation Context ...and context of these derivatives of the xi function. In particular, we briefly elaborate on important earlier numerical calculations. Proof of Proposition 1 The result (1) uses estimates of Grosswald =-=[9, 10]-=- and Hayman [13] and the notation here essentially follows that of Grosswald. The function Ξ(t) isevenand the coefficients of t2 alternate in sign. Putting ∞� (3) ξ(1/2+it) ≡ Ξ(t)= cnt n = f(−t 2 ∞� )... |

3 |
On a class of Fourier transforms
- Wintner
- 1936
(Show Context)
Citation Context ...SYMPTOTIC ESTIMATE OF ξ (2n) (1/2) 5 weaker hypotheses than Grosswald. Pertinent results on the maximum modulus and Fourier transform of the Riemann xi function are given by Haviland [12] and Wintner =-=[19, 20]-=-. The authors of [6], while investigating the Turán inequalities, used the moments (20) where (21) Φ(t) ≡ Given the power series (22) ˆ bm ≡ � ∞ 0 t 2m Φ(t)dt, ∞� (2n 4 π 2 e 9t − 3n 2 πe 5t )exp(−n 2... |

2 |
Correction and completion of the paper “Generalization of a formula of
- Grosswald
- 1969
(Show Context)
Citation Context ...and context of these derivatives of the xi function. In particular, we briefly elaborate on important earlier numerical calculations. Proof of Proposition 1 The result (1) uses estimates of Grosswald =-=[9, 10]-=- and Hayman [13] and the notation here essentially follows that of Grosswald. The function Ξ(t) isevenand the coefficients of t2 alternate in sign. Putting ∞� (3) ξ(1/2+it) ≡ Ξ(t)= cnt n = f(−t 2 ∞� )... |

2 | The Riemann Ξ-function under repeated differentiation
- Ki
(Show Context)
Citation Context ...is separately proved from Proposition 1 and obviously subsumes the latter. Neither result is contingent upon the Riemann hypothesis. Proposition 2 is strong enough to imply a very recent result of Ki =-=[14]-=- that demonstrated another conjecture stated by Farmer and Rhoades. The asymptotic result (2) exhibits the property ξ (j) (1 − s)=(−1) jξ (j) (s). After describing the proofs of (1) and (2) we provide... |

1 |
New results on the Stieltjes constants: Exact and asymptotic evaluation
- Coffey
- 2006
(Show Context)
Citation Context ...ermits estimation of the growth of xi function derivatives at other locations based upon Proposition 1. For s = 1, one method to express cj is in terms of the well-known Stieltjes constants γk (e.g., =-=[2, 3]-=-). In particular, for (30), the values ξ (j) (1) (cf. Corollary 2) precisely enter the Li criterion for the validity of the Riemann hypothesis [1, 4]. Our Proposition 2 is strong enough to provide an ... |

1 |
New summation results for the Stieltjes constants
- Coffey
- 2006
(Show Context)
Citation Context ...ermits estimation of the growth of xi function derivatives at other locations based upon Proposition 1. For s = 1, one method to express cj is in terms of the well-known Stieltjes constants γk (e.g., =-=[2, 3]-=-). In particular, for (30), the values ξ (j) (1) (cf. Corollary 2) precisely enter the Li criterion for the validity of the Riemann hypothesis [1, 4]. Our Proposition 2 is strong enough to provide an ... |

1 |
On the asymptotic behaviour of the Riemann ξ-function
- Haviland
- 1945
(Show Context)
Citation Context ...ein they invokesASYMPTOTIC ESTIMATE OF ξ (2n) (1/2) 5 weaker hypotheses than Grosswald. Pertinent results on the maximum modulus and Fourier transform of the Riemann xi function are given by Haviland =-=[12]-=- and Wintner [19, 20]. The authors of [6], while investigating the Turán inequalities, used the moments (20) where (21) Φ(t) ≡ Given the power series (22) ˆ bm ≡ � ∞ 0 t 2m Φ(t)dt, ∞� (2n 4 π 2 e 9t −... |

1 |
A detailed numerical examination of the tracking of the zeros of Fλ(z) to produce a lower bound for the de Bruijn-Newman constant Λ, ICM9011-04 technical report
- Norfolk, Ruttan, et al.
- 1990
(Show Context)
Citation Context ...inimum value after the initial decrease. Indeed, the initial decrease of ξ (j) (1/2) implies the monotonic decrease of the moments ˆ bm until m = 339. Based upon the minimum numerical value of ˆ b339 =-=[18, 16]-=-, we know that approximately ξ (678) (1/2) � 2.19259386×10 134 . The numerical result of Kreminski [15] agrees with this value. It turns out that all of the moments { ˆbm} 1000 m=0 were numerically de... |

1 |
A note on the Riemann ξ-function
- Wintner
- 1935
(Show Context)
Citation Context ...SYMPTOTIC ESTIMATE OF ξ (2n) (1/2) 5 weaker hypotheses than Grosswald. Pertinent results on the maximum modulus and Fourier transform of the Riemann xi function are given by Haviland [12] and Wintner =-=[19, 20]-=-. The authors of [6], while investigating the Turán inequalities, used the moments (20) where (21) Φ(t) ≡ Given the power series (22) ˆ bm ≡ � ∞ 0 t 2m Φ(t)dt, ∞� (2n 4 π 2 e 9t − 3n 2 πe 5t )exp(−n 2... |