# Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications

@inproceedings{Kaneko2021MotohashisFF, title={Motohashi's Formula for the Fourth Moment of Individual Dirichlet \$L\$-Functions and Applications}, author={Ikuya Kaneko}, year={2021} }

A new reciprocity formula for Dirichlet !-functions associated to an arbitrary primitive Dirichlet character of prime modulus @ is established. We find an identity relating the fourth moment of individual Dirichlet !-functions in the C-aspect to the cubic moment of central !-values of Hecke–Maaß newforms of level at most @ and primitive central character k averaged over all primitive nonquadratic characters kmodulo @. Our formulæ would be viewed as reverse versions of recent work of Petrow…

## One Citation

Spectral Moment Formulae for $GL(3)\times GL(2)$ $L$-functions

- Mathematics
- 2021

Spectral moment formulae of various shapes have proven to be very successful in studying the statistics of central L-values. In this article, we establish, in a completely explicit fashion, such…

## References

SHOWING 1-10 OF 77 REFERENCES

The Second Moment of the Product of the Riemann Zeta and Dirichlet $L$-Functions

- Mathematics
- 2021

We establish Motohashi’s formula for the second moment of the product of the Riemann zeta function and a Dirichlet !-function associated to an arbitrary primitive Dirichlet character modulo @ ∈ N. If…

Motohashi’s fourth moment identity for non-archimedean test functions and applications

- MathematicsCompositio Mathematica
- 2020

Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic $L$-functions. By an…

The Riemann Zeta-Function and Hecke Congruence Subgroups. II

- Mathematics
- 2007

This is a rework of our old file, which has been left unpublished since September 1994, on an explicit spectral decomposition of the fourth power moment of the Riemann zeta-function against a weight…

The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound.

- Mathematics
- 2019

We prove a Lindelof-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^*|d$, where $q^*$ is the…

A hybrid asymptotic formula for the second moment of Rankin–Selberg L-functions

- Mathematics
- 2012

Let g be a fixed modular form of full level, and let f j, k be a basis of holomorphic cuspidal newforms of even weight k, fixed level and fixed primitive nebentypus. We consider the Rankin-Selberg…

Twelfth moment of Dirichlet L-functions to prime power moduli

- Mathematics
- 2019

We prove the q-aspect analogue of Heath-Brown's result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. Our results rely on the p-adic…

Eisenstein series and the cubic moment for PGL(2)

- Mathematics
- 2019

Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized diagonal periods of products of Eisenstein…

On Motohashi's formula

- Mathematics
- 2020

We offer a new pespective of the proof of a Motohashi-type formula relating the fourth moment of $L$-functions for $\mathrm{GL}_1$ with the third moment of $L$-functions for $\mathrm{GL}_2$ over…

LEVEL RECIPROCITY IN THE TWISTED SECOND MOMENT OF RANKIN–SELBERG $L$ -FUNCTIONS

- Mathematics
- 2018

We prove an exact formula for the second moment of Rankin–Selberg $L$
-functions $L(\frac{1}{2},f\times g)$
twisted by $\unicode[STIX]{x1D706}_{f}(p)$
, where $g$
is a fixed holomorphic cusp form…

An explicit formula for the fourth power mean of the Riemann zeta-function

- Mathematics
- 1993

In the celebrated paper [1] Atkinson exhibited an explicit formula for the mean square of the Riemann zeta-function on the critical line, which greatly enriched the theory of this most fundamental…