Rankin–Selberg convolution ( $L (f, g, s)$ )

f = \sum a_{n} q^{n}

and

g = \sum b_{n} q^{n}

, their Rankin–Selberg convolution in the elementary normalization is

L (f, g, s) = \sum_{n \geq 1} a_{n} \overline{b_{n}} n^{- s}

.

Table of contents
- Rankin–Selberg unfolding identity for a holomorphic Eisenstein series Rankin–Selberg convolution

Rankin–Selberg unfolding identity for a holomorphic Eisenstein series

If

f

and

g

have respective weights

k

and

l

, and

r = k - l > 2

, unfolding the weight-

r

Eisenstein series gives

⟨ f, g G_{r} ⟩ = \frac{2 ζ ( r ) Γ ( k - 1 )}{( 4 π ) ^{k - 1}} L (f, g, k - 1) .

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 137 / 2 / c / Solution

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