The Second Integral Moment Of Maass L-functions

A crucial problem in the theory of automorphic L-functions is to investigate their integral moments where f is an automorphic form for SLn(Z) andκis a positive integer. This problem has long been an important subject which has applications in many fields. Over the past years, many mathematicians have worked in this field with fruitful results; see, for example, an excellent paper [17].In the case of GL1, this is to estimate the integral moments of Riemannζfunction and Dirichlet L-functions. Denote andFor Riemannζfunction, whenκ= 1, Hardy and Littlewood [6] proved the famous asymptotic formula The caseκ= 2 was considered by Ingham [10] and he derived Besides, for any positive integer k, Ramachandra [25] obtained the lower bound of I(κ,T), i.e. Later, Heath-Brown [7] showed the same bound for all positive rationalκ.For Dirichlet L-functions, Rane [27] proved the second integral moment But for fourth power moment, no asymtotic result has been obtained.Under the generalized Lindelof hypothesis, the estimates and are true for an arbitrary positive integer k. While forκ≥3, (5) and (6) have not been proved unconditionally. Heath-Brown [7] showed the following upper bound Meurman [19] derived There are other mean value results for Riemannζfunction and Dirichlet L-functions; see [3], [8], [9], [11], [20], [22], [24], [26], [30], and [31].In this paper, we will study the second integral mean value of L(s,f), where f is a normalized Hecke-Maass form for SL3(Z). Under the Ramanujan Conjecture, Matsumoto [18] obtained that for(?) where a1,n is the Fourier coefficient of f andε> 0 is arbitrary.Using a different approach, we can obtain the following unconditional result.Theorem 1.1 Let f be a normalized Hecke-Maass form of type (ν1,ν2) for SL3(Z). If (?), then we have where and a1,n is the Fourier coefficient of f.We will use the idea in [13] to get the approximate functional equation for L(s,f), i.e. representing L(s,f) as the sum of two parts. Each part is an essential finite summation. Then we estimate the two integral moment separately. We can prove that one part produces the main term, and the other part only contributes to the error term.Using the same method, we also consider the second integer moment of L'(s,f) and obtain the following result.Theorem 1.2 Under the same assumption of Theorem 1.1, we get where...