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1: Bibliography B

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More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

Encodings:

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Bernoulli numbers and polynomials of arbitrary complex indices. Appl. Math. Lett. 5 (6), pp. 83–88.

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External Links:

ISSN 0893-9659, MathReview, ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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External Links:

ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

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Euler functions $E_{\alpha }(z)$ with complex $\alpha$ and applications. In Approximation, probability, and related fields (Santa Barbara, CA, 1993), G. Anastassiou and S. T. Rachev (Eds.), pp. 127–150.

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External Links:

MathReview (R. N. Kalia), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Josef Meixner: his life and his orthogonal polynomials. Indag. Math. (N.S.) 30 (1), pp. 250–264.

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External Links:

ISSN 0019-3577, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

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2: 25.10 Zeros

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

3: Bibliography H

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A Boole-type Formula involving Conjugate Euler Polynomials. In Charlemagne and his Heritage. 1200 Years of Civilization and Science in Europe, Vol. 2 (Aachen, 1995), P.L. Butzer, H. Th. Jongen, and W. Oberschelp (Eds.), pp. 361–375.

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External Links:

ISBN 2-503-50674-7, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.

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Notes:

Reprinted in 1979.

External Links:

ISBN 0-8218-1556-3, MathReview, ZentralBlatt

Referenced by:

§35.4(i).

Encodings:

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4: 25.18 Methods of Computation

… ►See also Allasia and Besenghi (1989), Butzer and Hauss (1992), Kerimov (1980), and Yeremin et al. (1985). …

5: 24.16 Generalizations

… ►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

6: 19.37 Tables

… ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $k^2=0(.01)1$ to 10D by Fettis and Caslin (1964). ►Tabulated for $\varphi =0(1^{\circ }){90}^{\circ }$, $k^2=0(.01)1$ to 7S by Beli͡akov et al. (1962). … ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $k=0(.01)1$ to 10D by Fettis and Caslin (1964). ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971), for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(2^{\circ }){90}^{\circ }$ and $5^{\circ }({10}^{\circ }){85}^{\circ }$ to 8D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({10}^{\circ }){90}^{\circ }$, $\arcsin k=0(5^{\circ }){90}^{\circ }$ to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated (with different notation) for $\varphi =0({15}^{\circ }){90}^{\circ }$, ${\alpha }^2=0(.1)1$, $\arcsin k=0({15}^{\circ }){90}^{\circ }$ to 5D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({15}^{\circ }){90}^{\circ }$, ${\alpha }^2=0(.1)1$, $\arcsin k=0({15}^{\circ }){90}^{\circ }$ to 7D by Zhang and Jin (1996, pp. 676–677). …

7: 35.4 Partitions and Zonal Polynomials

… ►See Hua (1963, p. 30), Constantine (1963), James (1964), and Macdonald (1995, pp. 425–431) for further information on (35.4.2) and (35.4.3). …

8: Errata

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In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

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9: 20.15 Tables

… ►This reference gives ${\theta }_j(x,q)$, $j=1,2,3,4$, and their logarithmic $x$-derivatives to 4D for $x/\pi =0(.1)1$, $\alpha =0(9^{\circ }){90}^{\circ }$, where $\alpha$ is the modular angle given by … ►Spenceley and Spenceley (1947) tabulates ${\theta }_1(x,q)/{\theta }_2(0,q)$, ${\theta }_2(x,q)/{\theta }_2(0,q)$, ${\theta }_3(x,q)/{\theta }_4(0,q)$, ${\theta }_4(x,q)/{\theta }_4(0,q)$ to 12D for $u=0(1^{\circ }){90}^{\circ }$, $\alpha =0(1^{\circ }){89}^{\circ }$, where $u=2x/(\pi {\theta }_3^2(0,q))$ and $\alpha$ is defined by (20.15.1), together with the corresponding values of ${\theta }_2(0,q)$ and ${\theta }_4(0,q)$. ►Lawden (1989, pp. 270–279) tabulates ${\theta }_j(x,q)$, $j=1,2,3,4$, to 5D for $x=0(1^{\circ }){90}^{\circ }$, $q=0.1(.1)0.9$, and also $q$ to 5D for $k^2=0(.01)1$. ►Tables of Neville’s theta functions ${\theta }_s(x,q)$, ${\theta }_c(x,q)$, ${\theta }_d(x,q)$, ${\theta }_n(x,q)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\epsilon ,\alpha =0(5^{\circ }){90}^{\circ }$, where (in radian measure) $\epsilon =x/{\theta }_3^2(0,q)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …

10: 5.16 Sums

… ►For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).