Hecke eigenforms of degree two: Dirichlet series and Euler products

Our objective is to prove that certain Dirichlet series (in our variable q−s), which are defined by infinite sums, can be expressed as a product of an explicit rational function in q−s times an unknown polynomial M in q−s Moreover we show that M(q−s) is 1 if a simple condition is met. The Dirichlet series appear in the Euler products of Fourier coefficients for Eisenstein series. The series discussed below generalize the functions α0(N, q−s) used by Shimura in [12], and the theorem is an extension of Kitaoka’s result [5].

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Non-vanishing of Dirichlet series without Euler products

10.46298/hrj.2018.4027 ◽

2018 ◽

Vol Volume 40 ◽

Author(s):

William D. Banks

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Half Plane ◽

Riemann Zeta Function ◽

Euler Product ◽

The Riemann Zeta Function ◽

International Audience ◽

Euler Products ◽

Riemann Zeta

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

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On Ramanujan and Dirichlet series with Euler products

Glasgow Mathematical Journal ◽

10.1017/s0017089500005620 ◽

1984 ◽

Vol 25 (2) ◽

pp. 203-206 ◽

Cited By ~ 3

Author(s):

S. Raghavan

Keyword(s):

Modular Form ◽

Dirichlet Series ◽

Modular Forms ◽

Hecke Operators ◽

Euler Product ◽

Arithmetical Functions ◽

In Trans ◽

Euler Products ◽

Unpublished Manuscripts ◽

Profound Insight

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

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CONVERGENCE OF DIRICHLET SERIES AND EULER PRODUCTS

Contributions Section of Natural Mathematical and Biotechnical Sciences ◽

10.20903/csnmbs.masa.2017.38.2.111 ◽

2017 ◽

Vol 38 (2) ◽

pp. 153

Author(s):

Doug S. Phillips ◽

Peter Zvengrowski

Keyword(s):

Dirichlet Series ◽

Half Plane ◽

Convergence Theorem ◽

Sufficient Conditions ◽

Dirichlet Character ◽

Convergence Theorems ◽

Numerical Evidence ◽

Primitive Dirichlet Character ◽

Euler Products

The first part of this paper deals with Dirichlet series, and convergence theorems are proved that strengthen the classical convergence theorem as found e.g. in Serre’s “A Course in Arithmetic.” The second part deals with Euler-type products. A convergence theorem is proved giving sufficient conditions for such products to converge in the half-plane having real part greater than 1/2. Numerical evidence is also presented that suggests that the Euler products corresponding to Dirichlet L-functions L(s, χ), where χ is a primitive Dirichlet character, converge in this half-plane.

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The convergence of Euler products over p-adic number fields

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091507000636 ◽

2009 ◽

Vol 52 (3) ◽

pp. 583-606 ◽

Cited By ~ 1

Author(s):

Daniel Delbourgo

Keyword(s):

Topological Space ◽

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Number Fields ◽

Product Structure ◽

Euler Product ◽

Euler Products ◽

Riemann Zeta ◽

New Formula

AbstractWe define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler–Mascheroni constant.

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