The mean square of Dirichlet series associated with automorphic forms

Müller, Wolfgang

doi:10.1007/bf01303063

Cited by 28 publications

(25 citation statements)

References 11 publications

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“…Finally, the identities given in the theorem result from these expansions. As in we define

{normalΓ}_{α} false(s false) : = 2^{α} \frac{{normalΓ}^{2} false(s + 1 / 2 false)}{Γ (s + 1 - α)}_{2} F_{1} (1 / 2 - α, 1 / 2 - α; s + 1 - α; 1 / 2)

and set

F false(s + 1 / 2 false) : = \frac{1}{{normalΓ}^{2} false(s + 1 false)} (Γ_{3 / 4} (s + 1 / 2) - \frac{1}{4} Γ_{- 1 / 4} (s + 1 / 2)) .

Using the definition of

Γ_{α}

and a Gauß contiguous relation (see ), this expression can be simplified to

\begin{matrix} true \\ right & left F false(s + 1 / 2 false) \\ right & left = \frac{2^{3 / 4}}{normalΓ false(s + 3 / 4 false)} (()_{2} F_{1} (- 1 / 4, - 1 / 4; s + 3 / 4; 1 / 2) - \frac{1}{8 (s + 3 / 4)}_{2} F_{1} (3 / 4, 3 / 4; s + 7 / 4; 1 / 2)) \\ left = \frac{2^{3 / 4}}{normalΓ false(s + 3 / 4 false)}_{2} F_{1} (() ...) \end{matrix}

…”

Section: Generalised Dirichlet L‐functionsmentioning

confidence: 99%

“…This function appears as the Mellin transform of the Whittaker function. According to [, Equation 52], the Laurent expansion of

M (f, s + 1 / 2)

about 0 is

M false(f, s + 1 / 2 false) = \frac{1}{2^{5 / 2} s^{2}} + \frac{2 {\overset{a}{true}}_{0} - \log 2}{2^{3 / 2} s} + O false(1 false) .

Moreover, the Laurent expansion of

M (E_{1 / 2} f, s + 1 / 2)

about

0

equals

M false(E_{1 / 2} f, s + 1 / 2 false) = \frac{1}{2^{9 / 2} s^{2}} + \frac{1}{2^{7 / 2} s} (2 {\hat{a}}_{0} - \log 2 + 2) + O false(1 false) .

Here

E_{1 / 2}

denotes the Maaß lowering operator which is given by

E_{1 / 2} = y (i \frac{\partial}{\partial x} - \frac{\partial}{\partial y}) + \frac{1}{4}

The Laurent expansions of

M (f, - (s + 1 / 2))

and

M (E_{1 / 2} f, - false(s + 1 / 2 false))

about

0

are given by …”

Section: Generalised Dirichlet L‐functionsmentioning

confidence: 99%

See 1 more Smart Citation

Mixed moment of GL(2) and GL(3)L‐functions

Balkanova

Bhowmik

Frolenkov

et al. 2019

Proc. London Math. Soc.

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Let frakturf run over the set H4k of primitive cusp forms of level one and weight 4k, k∈N. We prove an explicit formula for the mixed moment of the Hecke L‐function L(f,1/2) and the symmetric square L‐function L(prefixsym2f,1/2), relating it to the dual mixed moment of a double Dirichlet series and the Riemann zeta function weighted by the 3F2 hypergeometric function. Analysing the corresponding special functions by means of the Liouville–Green approximation followed by the saddle point method, we prove that the initial mixed moment is bounded above by log3k.

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“…Finally, the identities given in the theorem result from these expansions. As in we define

{normalΓ}_{α} false(s false) : = 2^{α} \frac{{normalΓ}^{2} false(s + 1 / 2 false)}{Γ (s + 1 - α)}_{2} F_{1} (1 / 2 - α, 1 / 2 - α; s + 1 - α; 1 / 2)

and set

F false(s + 1 / 2 false) : = \frac{1}{{normalΓ}^{2} false(s + 1 false)} (Γ_{3 / 4} (s + 1 / 2) - \frac{1}{4} Γ_{- 1 / 4} (s + 1 / 2)) .

Using the definition of

Γ_{α}

and a Gauß contiguous relation (see ), this expression can be simplified to

\begin{matrix} true \\ right & left F false(s + 1 / 2 false) \\ right & left = \frac{2^{3 / 4}}{normalΓ false(s + 3 / 4 false)} (()_{2} F_{1} (- 1 / 4, - 1 / 4; s + 3 / 4; 1 / 2) - \frac{1}{8 (s + 3 / 4)}_{2} F_{1} (3 / 4, 3 / 4; s + 7 / 4; 1 / 2)) \\ left = \frac{2^{3 / 4}}{normalΓ false(s + 3 / 4 false)}_{2} F_{1} (() ...) \end{matrix}

…”

Section: Generalised Dirichlet L‐functionsmentioning

confidence: 99%

“…This function appears as the Mellin transform of the Whittaker function. According to [, Equation 52], the Laurent expansion of

M (f, s + 1 / 2)

about 0 is

M false(f, s + 1 / 2 false) = \frac{1}{2^{5 / 2} s^{2}} + \frac{2 {\overset{a}{true}}_{0} - \log 2}{2^{3 / 2} s} + O false(1 false) .

Moreover, the Laurent expansion of

M (E_{1 / 2} f, s + 1 / 2)

about

0

equals

M false(E_{1 / 2} f, s + 1 / 2 false) = \frac{1}{2^{9 / 2} s^{2}} + \frac{1}{2^{7 / 2} s} (2 {\hat{a}}_{0} - \log 2 + 2) + O false(1 false) .

Here

E_{1 / 2}

denotes the Maaß lowering operator which is given by

E_{1 / 2} = y (i \frac{\partial}{\partial x} - \frac{\partial}{\partial y}) + \frac{1}{4}

The Laurent expansions of

M (f, - (s + 1 / 2))

and

M (E_{1 / 2} f, - false(s + 1 / 2 false))

about

0

are given by …”

Section: Generalised Dirichlet L‐functionsmentioning

confidence: 99%

Mixed moment of GL(2) and GL(3)L‐functions

Balkanova

Bhowmik

Frolenkov

et al. 2019

Proc. London Math. Soc.

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show abstract

“…This statement can be proved using the properties of the Rankin-Selberg convolution or from the application of the circle method to study the asymptotics of |θ(z)| 6 dz [1,3] (see also [7] for a general theorem). 2…”

Section: Auxiliary Lemmatamentioning

confidence: 99%

Visible lattice points in the sphere

Chamizo

Cristóbal

Ubis

2007

Journal of Number Theory

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The number of visible (primitive) lattice points in the sphere of radius R is well approximated by 4π 3ζ(3) R 3 . We consider an integral expression involving the error term E * (R), which leads to E * (R) = Ω(R(log R) 1/2 ). This is comparable to what is known in the sphere problem. We can avoid the use of the second power moment (which is in this case unknown) by employing an auxiliary trigonometric series correlated to E * (R). This approach to prove Ω-results seems to be new and could be useful in other problems.

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“…For details regarding discontinuous groups and automorphic forms, see [8,10,11,14,15,16]. Let H = {z ∈ C : (z) > 0} denote the upper half plane and G = S L(2, R) the special linear group of all 2 × 2 matrices with determinant 1.…”

Section: S P  Tmentioning

confidence: 99%

“…Zagier [20] extended the method to cover forms that are not cuspidal and may not decay rapidly at infinity. Müller [11,12] considered the case where a(n) is the Fourier coefficient of non-holomorphic cusp or non-cusp form of real weight with respect to a Fuchsian group of the first kind. It is this last approach we wish to discuss.…”

Section: S P  Tmentioning

confidence: 99%