An Explicit Upper Bound for the Riemann Zeta-Function near the Line $\sigma=1$

Cheng, Yuanyou Furui

doi:10.1216/rmjm/1181071682

Cited by 14 publications

(11 citation statements)

References 9 publications

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“…This is to compare to Proposition 1 of Cheng [2] who obtained 5.505 instead of 2.1946t −σ . When σ ≥ 1/2 and 0 ≤ t ≤ 15, a Mathematica computation gives us that |ζ(s) − 1≤n<t n −s | ≤ 43t −σ .…”

Section: Introductionsupporting

confidence: 59%

A zero density result for the Riemann zeta function

Kadiri

2013

Acta Arith.

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Section: Introductionsupporting

confidence: 59%

A zero density result for the Riemann zeta function

Kadiri

2013

Acta Arith.

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“…as in [3]. We also use the trigonometric identities sin a sin b 1 2 cosa À b À cosa b; sin a cos b 1 2 sina b sina À b:…”

Section: Theorem 2 For Small Lmentioning

confidence: 99%

“…as in [3]. We also use the trigonometric identities To bound I m , let f x x À j =t 6 2pmx and gx kt log x 6 2pmx, where k H x hx and kx P f6 sinx; 6 cosxg.…”

Section: Theorem 2 For Small Lmentioning

confidence: 99%

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Vinogradov's Integral and Bounds for the Riemann Zeta Function

Ford

2002

Proc. Lond. Math. Soc.

127

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kevin ford 1 < x < P ea 1 x . . . a k x k 2 s eÀa 1 h 1 À . . . À a k h k da < J s; k P; 0; . . . ; 0 J s; k P:Hence, writing Q P, we obtainAlso, counting only the solutions of (1.4) with x i y i for each i gives J s; k P > Q s . Therefore J s; k P > max2s À k P 2 s À kk 1 = 2 ; P s : 1:5 kevin fordUpper bounds take the form of J s; k P < Ds; kP 2 s À kk 1 = 2 hs; k ; 1:6where hs; k > 0 and Ds; k is independent of P. Stechkin in 1975 [24] proved (1.6) with hrk; k 1 2 k 2 1 À 1=k r ; Drk; k expfC minr; kk 2 log kg for an absolute constant C. The constant factor was improved by Wooley [31]. Small improvements to the exponents of P were subsequently made by Arkhipov and Karatsuba [1] and Tyrina [26] (signi®cant for s p k 2 ). Also signi®cant is Wooley's result [32] when s p k 3 = 2 À « , which is very close to the`ideal' bounds Ck; sP s in that range of s. For our purposes, the most important improvement comes from Wooley [30], who improved the exponents substantially in a wide range of s, showing that (1.6) holds with hk; s < 1 2 k 2 e 1 = 2 À 2 s = k 2 valid for s p k 2 log k (see [5, Lemma 5.2]). In Theorem 3 below, we combine Wooley's method with the main idea from [1] to improve this to hk; s < 3 8 k 2 e 1 = 2 À 2 s = k 2 . In the application to bounding the Riemann zeta function, we will take s to be of order k 2 , so this small improvement is signi®cant.Theorem 3. Let k and s be integers with k > 1000 and 2k 2 < s < 1 2 k 2 1 2 log 3 8 k. Then J s; k P < k 2:055 k 3 À 5:91 k 2 3 s 1:06 sk 2 s 2 = k À 9:7278 k 3 P 2 s À kk 1 = 2 D s P > 1;where D s 3 8 k 2 e 1 = 2 À 2 s = k 2 1:7 = k : Further, if k > 129, there is an integer s < rk 2 such that for P > 1, J s; k P < k vk 3 P 2 s À kk 1 = 2 0:001 k 2

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“…If we assume c 4 c 0 = 10 8 , this leads to σ 0 = 14/13. Strengthening the above approach to the Lagarias problem requires the derivation of a good explicit value for the constant c 1 (compare [3]) and knowledge of the best current value value for c 0 (currently 3.2 × 10 9 ) [12].…”

mentioning

confidence: 99%

Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip

Broughan

2009

Can. math. bull.

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Abstract. If K is a number field with n k = [k : Q], and ξ k the symmetrized Dedekind zeta function of the field, the inequalityfor t = 0 is shown to be true for σ ≥ 1 + 8/n 1 3 k improving the result of Lagarias where the constant in the inequality was 9. In the case k = Q the inequality is extended to σ ≥ 1 for all t sufficiently large or small and to the region σ ≥ 1 + 1/(log t − 5) for all t = 0. This answers positively a question posed

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An Explicit Upper Bound for the Riemann Zeta-Function near the Line $\sigma=1$

Cited by 14 publications

References 9 publications

A zero density result for the Riemann zeta function

A zero density result for the Riemann zeta function

Vinogradov's Integral and Bounds for the Riemann Zeta Function

Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip

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