The sum of like powers of the zeros of the Riemann zeta function

Lehmer, D. H.

doi:10.1090/s0025-5718-1988-0917834-x

Cited by 18 publications

(20 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One reason for the importance of the coefficients σ k is their correspondence to sums of reciprocal powers of the nontrivial zeros ρ of the ζ function [24,31]: (4) we see that the connection between the values λ n and the sequence {σ k } is λ n = − n j=1 (−1) j n j σ j . Further discussion of the σ j 's is presented in Appendix J.…”

Section: Estimation Of Sumsmentioning

confidence: 99%

Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

Coffey

2005

Math Phys Anal Geom

View full text Add to dashboard Cite

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence {λ k }, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence η j are valid. The constants η j enter the Laurent expansion of the logarithmic derivative of the zeta function about s = 1 and appear to have remarkable characteristics. On our conjecture, not only does the Riemann hypothesis follow, but an inequality governing the values λ n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.

show abstract

Section: Estimation Of Sumsmentioning

confidence: 99%

Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

Coffey

2005

Math Phys Anal Geom

View full text Add to dashboard Cite

show abstract

“…Other sums related to the Z(k) are Z j = ρ ρ −j [22,16,26] (often denoted σ j , but here we use σ as variable). It was already known that λ n = n j=1 (−1) j+1 n j Z j [12, Equation (27)], and that the Z j in turn are complicated polynomials in the Stieltjes constants {γ k } k<j [22] (for λ n and γ k see also [21,3,4] and references therein).…”

Section: Background and Notationsmentioning

confidence: 99%

“…A related but more concrete requirement can be put on the function N(T ), the number of zeros of L(s) with 0 < Im ρ < T : we ask that for some constants R −2 , R −1 and some α < 1 (all now depending on the chosen L-series), (16) (implying R −2 ≥ 0). If both conditions (5) and (16) hold, then the polar coefficients of Z(σ) in (5) have to be precisely the R −j from (16). All of that is realized in many cases including, but not limited to, Dedekind zeta functions [15,13,27] and some Dirichlet L-functions [5,27,18]; see also [14] (discussed at end); in all those instances, δN(T ) = O(log T ).…”

Section: [Rh True]mentioning

confidence: 99%

Sharpenings of Li's Criterion for the Riemann Hypothesis

Voros

2006

Math Phys Anal Geom

View full text Add to dashboard Cite

Exact and asymptotic formulae are displayed for the coefficients λ n used in Li's criterion for the Riemann Hypothesis. For n → ∞ we obtain that if (and only if) the Hypothesis is true, λ n ∼ n(A log n + B) (with A > 0 and B explicitly given, also for the case of more general zeta or L-functions); whereas in the opposite case, λ n has a non-tempered oscillatory form.Li's criterion for the Riemann Hypothesis (RH) states that the latter is true if and only if a specific real sequence {λ n } n=1,2,... has all its terms positive [17, 2]. Here we show that it actually suffices to probe the λ n for their large-n behavior, which fully encodes the Riemann Hypothesis by way of a clear-cut and explicit asymptotic alternative. To wit, we first represent λ n exactly by a finite oscillatory sum (9), then by a derived integral formula (12), which can finally be evaluated by the saddle-point method in the n → +∞ limit. As a result, λ n takes one of two sharply distinct and mutually exclusive asymptotic forms: if RH is true, λ n will grow tamely according to (17); if RH is false, λ n will oscillate with an exponentially growing amplitude, in both + and − directions, as described by (18). This dichotomy thus provides a sharp criterion of a new asymptotic type for the Riemann Hypothesis (and for other zeta-type functions as well, replacing (17) by (15)).This work basically reexposes our results of April 2004 announced in [28], but with an uncompressed text; we also update the references and related comments: for instance, we now derive as (24) a large-n expansion surmised by Maślanka [20] in the meantime.

show abstract

“…The third stream is concerned with relations between the coefficients of power series of ξ and its logarithm and sums over inverse powers of the zeros of ξ [9,6,10,11]. These relations enable necessary conditions to be established for the Riemann hypothesis to hold, and to do this use a mapping from the critical line to the unit circle.…”

Section: Introductionmentioning

confidence: 99%

Towards a resolution of the Riemann hypothesis

McPhedran

2020

Preprint

View full text Add to dashboard Cite

This article contains work associated with a resolution of the Riemann hypothesis, following work by Taylor [1], Lagarias and Suzuki [2] and Ki [3], as well as Pustyl'nikov [4,5] and Keiper [6]. Functions ξ+(s) and ξ−(s) are considered, for which it is known that all zeros lie on the critical line. The Riemann hypothesis itself pertains to the question of the location of the zeros of the sum and difference of ξ+(s) and ξ−(s), and this is investigated. An argument is developed which prima facie establishes the validity of the Riemann hypothesis. It adds to a necessary condition of Pustyl'nikov a sufficient condition. A second argument is discussed, which could have been accessible to Riemann.

show abstract

The sum of like powers of the zeros of the Riemann zeta function

Cited by 18 publications

References 4 publications

Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

Toward Verification of the Riemann Hypothesis: Application of the Li Criterion

Sharpenings of Li's Criterion for the Riemann Hypothesis

Towards a resolution of the Riemann hypothesis

Contact Info

Product

Resources

About