Formulas for Higher Derivatives of the Riemann Zeta Function

Apostol, Tom M.

doi:10.2307/2007806

Mathematics of Computation

1985

DOI: 10.2307/2007806

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Formulas for Higher Derivatives of the Riemann Zeta Function

Tom M. Apostol¹

Abstract: Abstract. The functional equation for f(i) is used to obtain formulas for all derivatives f(Ar)(î). A closed form evaluation of f(Ar)(0) is given, and numerical values are computed to 15D for k = 0(1)18.

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1988

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A note on 𝜁”(𝑠) and 𝜁”’(𝑠)

Yıldırım

1996

Proc. Amer. Math. Soc.

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Abstract. There is only one pair of non-real zeros of ζ (s), and of ζ (s), in the left half-plane. The Riemann Hypothesis implies that ζ (s) and ζ (s) have no zeros in the strip 0 ≤ s < It was shown by Speiser [3] that the Riemann Hypothesis is equivalent to ζ (s) having no zeros in 0 < σ < 1 2 (as usual we write s = σ + it). Levinson and Montgomery [2] gave a different proof; moreover, they showed that ζ (k) (s) has at most a finite number of non-real zeros in σ <

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A note on 𝜁”(𝑠) and 𝜁”’(𝑠)

Yıldırım

1996

Proc. Amer. Math. Soc.

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

Lehmer¹

1988

Math. Comp.

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Abstract.In this paper we discuss a method of evaluating the sum cr = ^2 p~T where r is an integer greater than 1 and the sum is taken over all the complex zeros of c (s), the Riemann zeta function. The method requires the coefficients of the Maclaurin expansion of the entire function f(s) = (s -l)f(s). These are obtained from a limit theorem of Sitaramachandrarao by the use of the Euler-Maclaurin summation formula. The sum <7r is then obtained from the logarithmic derivative of the function f(s). A table of o> is given to 30 decimals for r = 2 (1)

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The Stieltjes function—definition and properties

Bohman¹,

Fröberg²

1988

Math. Comp.

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Formulas for Higher Derivatives of the Riemann Zeta Function

Abstract: Abstract. The functional equation for f(i) is used to obtain formulas for all derivatives f(Ar)(î). A closed form evaluation of f(Ar)(0) is given, and numerical values are computed to 15D for k = 0(1)18.

Cited by 5 publications

References 2 publications

A note on 𝜁”(𝑠) and 𝜁”’(𝑠)

A note on 𝜁”(𝑠) and 𝜁”’(𝑠)

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

The Stieltjes function—definition and properties

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