The fourth moment of individual Dirichlet L -functions on the critical line
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Mathematische Zeitschrift
The fourth moment of individual Dirichlet L-functions on the critical line Berke Topacogullari1 Received: 29 October 2019 / Accepted: 21 August 2020 © The Author(s) 2020
Abstract We prove an asymptotic formula for the second moment of a product of two Dirichlet Lfunctions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line. Keywords Moments of L-functions · Dirichlet L-functions · Dedekind zeta functions Mathematics Subject Classification 11M06
1 Introduction Moments of L-functions are a central topic in analytic number theory, not only due to their many important applications, but also because they give insight into the behaviour of Lfunctions in the critical strip. One of the most famous and best-studied examples in this regard is the fourth moment of the Riemann zeta function T 1 ζ + it 4 dt. (1.1) 2 1
The first asymptotic formula for (1.1) goes back to Ingham [22], who proved that T 1 ζ + it 4 dt = 1 T (log T )4 + O T (log T )3 . 2 2 2π 1 It was not until several decades later that Heath-Brown [18] was able to improve on this estimate. His result, which marked a major advance in the subject, states that T 1 ζ + it 4 dt = T P(log T ) + O T 78 +ε , (1.2) 2 1
B 1
Berke Topacogullari [email protected] EPFL SB MATH TAN, Station 8, 1015 Lausanne, Switzerland
123
B. Topacogullari
where P is a certain polynomial of degree 4. Further progress came with the development of methods originating in the spectral theory of automorphic forms, in particular the Kuznetsov formula [32]. Zavorotnyi [49] was thus able to lower the exponent in the error term in (1.2) and show that
T
ζ
1
1 2
2 4 + it dt = T P(log T ) + O T 3 +ε .
(1.3)
Motohashi [38, Theorem 4.2] established an explicit formula which expresses a smooth version of the fourth moment (1.1) in terms of the cubes of the central values of certain automorphic L-functions. His result is significant, as it allows a much deeper understanding of (1.1) than a mere asymptotic estimate, in addition to having many remarkable applications (see e.g. [24,25]). The best estimate for (1.1) to date is due to Ivi´c and Motohashi [25, Theorem 1] who, by making use of the explicit formula, were able to replace the factor T ε in (1.3) by a suitable power of log T . In this article, we are interested in the analogous problem for Dirichlet L-functions. Naturally, the fourth moment can here be taken in two different ways: on the one hand, we can look at an individual Dirichlet L-function and take the average along the critical line as in (1.1). On the other hand, we can focus on the central point s = 1/2 and take the average over a suitable subset of Dirichlet characters, most typically the set of all
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