Nullstellen Epsteinscher Zetafunktionen

Kober, H.

doi:10.1112/plms/s2-42.1.1

Cited by 2 publications

(6 citation statements)

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“…Littlewood [46,59,80], as well as later adaptations to other Dirichlet series by Kober [73] and Hecke [62], the (1.20)…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…For the classical Epstein zeta function Z 2 (s, Q), this fact was proved for the first time by Potter and Titchmarsh [95]. A simpler proof, similar to Hardy's proof of the same theorem for ζ (s), was given by Kober [73].…”

Section: The Zeros Of a Class Of Dirichlet Seriesmentioning

confidence: 86%

“…In the previous two corollaries we gave conditions assuring the infinitude of zeros at the critical line of a large class of Dirichlet series. As particular cases, these include large subclasses of the Epstein zeta function and the Dedekind zeta function attached to an imaginary quadratic field [30,73]. We may apply the previous result for the non-diagonal Epstein zeta function studied in the previous section, which satisfies Hecke's functional equation with r = 1.…”

Section: The Zeros Of a Class Of Dirichlet Seriesmentioning

confidence: 96%

“…Kober [73] established the condition given in Corollary 4.2 for a class of Epstein zeta functions attached to binary and positive definite quadratic forms whenever a/ |d| is irrational. For the case where a/ |d| is rational, then by using a set of conditions similar to that given in Corollary 4.3, we are only left with a finite set of Epstein zeta functions for which we need to verify Hardy's Theorem (see [73], p. 8).…”

Section: 26)mentioning

confidence: 99%

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On the Epstein zeta function and the zeros of a class of Dirichlet series

Ribeiro¹,

Yakubovich²

2021

Preprint

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By generalizing the classical Selberg-Chowla formula, we establish the analytic continuation and functional equation for a large class of Epstein zeta functions. This continuation is studied in order to provide new classes of theorems regarding the distribution of zeros of Dirichlet series in their critical lines. Due to the symmetries provided by the representation via the Selberg-Chowla formula, some generalizations of well-known formulas in analytic number theory are given as examples.

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“…Littlewood [46,59,80], as well as later adaptations to other Dirichlet series by Kober [73] and Hecke [62], the (1.20)…”

Section: Preliminary Resultsmentioning

confidence: 99%

Section: The Zeros Of a Class Of Dirichlet Seriesmentioning

confidence: 86%

Section: The Zeros Of a Class Of Dirichlet Seriesmentioning

confidence: 96%

Section: 26)mentioning

confidence: 99%

See 2 more Smart Citations

On the Epstein zeta function and the zeros of a class of Dirichlet series

Ribeiro¹,

Yakubovich²

2021

Preprint

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“…Kober denotes the right-hand side of (6.3) by Q(u, v, w) and comments that it is a regular function of w, apart from the poles at wZ1/2 and wZK1/2. Note that the Macdonald function satisfies K w ðzÞZ K Kw ðzÞ, so that the sum on the right-hand side of (6.3) is even in w. It has been shown by Potter & Titchmarsh (1935) and also by Kober (1936) that Z(a, b, c; s) has an infinity of zeros on the critical line Re(s)O1/2. Potter & Titchmarsh (1935) also exhibited numerically two zeros of Z(1, 0, 5; s) not lying on the critical line.…”

Section: A Two-dimensional Lattice Sum and Its Bessel Function Representationmentioning

confidence: 99%

On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums

et al. 2008

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We consider a general class of two-dimensional lattice sums consisting of complex powers s of inverse quadratic functions. We consider two cases, one where the quadratic function is negative definite and another more restricted case where it is positive definite. In the former, we use a representation due to H. Kober, and consider the limit u →∞, where the lattice becomes ever more elongated along one period direction (the one-dimensional limit). In the latter, we use an explicit evaluation of the sum due to Zucker and Robertson. In either case, we show that the one-dimensional limit of the sum is given in terms of ζ (2 s ) if Re( s )>1/2 and either ζ (2 s −1) or ζ (2−2 s ) if Re( s )<1/2. In either case, this leads to a Riemann property of these sums in the one-dimensional limit: their zeros must lie on the critical line Re( s )=1/2. We also comment on a class of sums that involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. We show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it; others are identically zero on it, while still others have no zeros on it.

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Nullstellen Epsteinscher Zetafunktionen

Cited by 2 publications

References 0 publications

On the Epstein zeta function and the zeros of a class of Dirichlet series

On the Epstein zeta function and the zeros of a class of Dirichlet series

On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums

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