An application of the projections of $C^{∞}$ automorphic forms

Noda, Takumi

doi:10.4064/aa-72-3-229-234

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…As a special case of Corollary 1, we obtain Noda's identity in [14] which relates the Fourier coefficients of the holomorphic cusp form f and the zeros of the Riemann zeta-function or the zeros of the symmetric square Lfunction of f . In addition, Theorem 1 gives an analytic series expansion of the central value L(1/2, f × g).…”

Section: It Satisfies the (Trivial) Functional Equation C(s) = C(1 − S)mentioning

confidence: 98%

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On zeros of approximate functions of the Rankin–Selberg L-functions

Suzuki

2009

Acta Arith.

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Notations. As usual, Z is the ring of rational integers, Z >0 the set of positive integers, C the field of complex numbers. We denote by h the upper half-plane, and by Γ the full modular group PSL 2 (Z). For a complex variable s, we put e(s) = e 2πis , Γ R (s) = π −s/2 Γ (s/2) and Γ C (s) = 2(2π) −s Γ (s). We denote by ζ(s) and ζ * (s) = Γ R (s)ζ(s) the Riemann zeta-function and the completed Riemann zeta-function, respectively, and denote by σ ν (n) = d|n d ν the divisor function. Throughout the paper, z = x + iy (x ∈ R, y > 0) is a variable on h, and s = σ + it (σ, t ∈ R) is a complex variable. A sum over the empty set is meant to be zero.

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Section: It Satisfies the (Trivial) Functional Equation C(s) = C(1 − S)mentioning

confidence: 98%

“…Further (5.2) holds for all s ∈ C, since f is a cusp form. By Lemma 1 of [14], the product f (z)E(z, s) is a C ∞ -modular form of bounded growth for 0 < σ < 1. Hence, by (4.4), we have…”

Section: 4mentioning

confidence: 99%

On zeros of approximate functions of the Rankin–Selberg L-functions

Suzuki

2009

Acta Arith.

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“…Our region − k 2 + 2 < Re(s) < k 2 − 1 includes the critical strip of Rankin zeta-functions. This is an advantage because it makes it possible to evaluate the zeta-functions at non-trivial zeros (see [6,7]). The proof of [6,Lemma 1] in fact is insufficient to imply the assertion, however the insufficiency is corrected essentially in the author's subsequent paper [7].…”

Section: Remarkmentioning

confidence: 99%

A note on the non-holomorphic Eisenstein series

Noda

2007

Ramanujan J

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We give a sufficient condition of bounded growth for the non-holomorphic Eisenstein series on SL 2 (Z). The C ∞ -automorphic forms of bounded growth are introduced by Sturm (Duke Math. J. 48 (2), 1981) in the study of automorphic L-functions. We also give a Laplace-Mellin transform of the Fourier coefficients of the Eisenstein series. The transformation constructs a projection of the Eisenstein series to the space of holomorphic cusp forms.

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On the zeros of symmetric square $L$-functions

Noda¹

1999

Kodai Math. J.

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An application of the projections of $C^{∞}$ automorphic forms

Cited by 4 publications

References 3 publications

On zeros of approximate functions of the Rankin–Selberg L-functions

On zeros of approximate functions of the Rankin–Selberg L-functions

A note on the non-holomorphic Eisenstein series

On the zeros of symmetric square $L$-functions

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