Spectral theory of the Riemann zeta-function.

*(English)*Zbl 0878.11001
Cambridge Tracts in Mathematics. 127. Cambridge: Cambridge University Press. ix, 228 p. (1997).

This important work brings forth the spectral theory of the Riemann zeta-function \(\zeta(s)\). This is a new, rich and rapidly developing field, and the present monograph gives its readers the current state-of-the-art of the subject. From the appearance of the Kuznetsov trace formulas some 20 years ago, Kloosterman sums and spectral theory have played an increasingly important rôle in analytic number theory. Now the author has succeeded in giving a comprehensive account of this theory, based to a large extent on his own research results. The main aim is to develop the theory of powerful new methods from spectral theory, and then to apply them to the study of \(\zeta(s)\), one of the classical and central objects of analytic number theory. This innovative spectral-theoretic approach, by no means exhausted yet, seems to have opened new perspectives in the theory of \(\zeta(s)\), which is noted for the depth and importance of its problems, in particular those concerning power moments and the distribution of the zeros. Spectral theory appears to be particularly well-suited in dealing with \(\int_0^T|\zeta(1/2+it)|^4 dt\), where it has provided exciting new results.

The content of the book is as follows. Chapter 1. Non-Euclidean harmonics. This is the introductory chapter, laying down the groundwork. The author proves the spectral resolution of the non-Euclidean Laplacian \[ \Delta = -y^2((\partial/\partial x)^2 + (\partial/\partial y)^2) \] with minimum prerequisites. Poincaré, Eisenstein series and Maass wave forms are introduced and their basic properties are developed. Chapter 2. Trace formulas. Transformation formulas for the sums \[ \sum_{\ell=1}^\infty {1\over\ell}S(m,\pm n;\ell)\varphi\left({4\pi\over\ell} \sqrt{mn}\right) \] are proved, where \(m,n\) are arbitrary positive integers, \(\varphi\) is a suitable smooth function and \(S(m,n;\ell)\) is the Kloosterman sum. The results are expressed in terms of spectral sums of Fourier coefficients of holomorphic and real-analytic cusp forms over the full modular group \(\Gamma\). These trace formulas play an important rôle in subsequent investigations of \(\zeta(s)\).

Chapter 3. Automorphic \(L\)–functions. Hecke operators and Hecke series \(H_j(s)\) are discussed. The theory is later used to obtain results on bilinear forms of Hecke \(L\)–functions and a non-vanishing result on \(H_j(1/2)\). Spectral large sieve results are obtained, and the important upper bound \[ \sum_{\kappa_j\leq K}\alpha_jH_j^4(1/2) \ll K^2\log^{15}K \] is derived. Here as usual \( \{\lambda_j = \kappa_j^2 + {1\over4}\} \cup \{0\} \) is the discrete spectrum of the non-Euclidean Laplacian and \(\alpha_j = |\rho_j(1)|^2(\cosh\pi\kappa_j)^{-1}\), where \(\rho_j(1)\) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue \(\lambda_j\) to which the Hecke series \(H_j(s)\) is attached.

Chapter 4. An explicit formula. The last two chapters of the book are devoted to the investigation of \(\zeta(s)\). The main object of study is the power moment \[ {\mathcal Z}_k(g) = \int_{-\infty}^\infty|\zeta(1/2+it)|^{2k}g(t) dt, \] where \(k\) is a given natural number, \(g(r)\) takes real values on the real axis and there exists a large positive constant \(A\) such that \(g(r)\) is regular and \(\ll (|r|+1)^{-A}\) for \(|\text{ Im }r|\leq A\). The cases \(k = 1,2\) are treated in detail, of which \(k = 2\) (the fourth moment) is of special importance. The tour de force is Theorem 4.2, which gives the explicit spectral decomposition of \({\mathcal Z}_2(g)\), with no error terms. This decomposition reveals a structural view of a beautiful relation between \(\zeta(s)\) and the entire family of Hecke \(L\)–functions. Although it is possible to develop a theory that will treat the zeta- and allied functions simultaneously, this is not done, since a general argument would fail to exploit the peculiar properties of \(\zeta(s)\) and, on the other hand, it would involve considerable technicalities which might cloud the main ideas.

Chapter 5. Asymptotics. In the last chapter the author chooses the Gaussian weight \[ g(t) = (\sqrt{\pi}G)^{-1}\exp(-(T-t)G^{-2}) \] and evaluates \({\mathcal Z}_2(g)\) using Theorem 4.2. For \(T^{1/2}\log^{-D}T \leq G \leq T/\log T\) and an arbitrary constant \(D > 0\) one has \[ \begin{split} {1\over\sqrt{\pi}G}\int_{-\infty}^\infty |\zeta(1/2+it+iT)|^4\exp(-(t/G)^2) dt\\ = {\pi\over\sqrt{2T}}\sum_{j=1}^\infty\alpha_jH_j^3(1/2)\kappa_j^{-1/2} \sin\left(\kappa_j\log{\kappa_j\over4eT}\right)\exp\left(-{1\over4} (G\kappa_j/T)^2\right) + O(\log^{3D+9}T).\end{split}(1) \] The asymptotic formula (1) is used then to obtain results on the error term \[ E_2(T) = \int_0^T|\zeta(1/2+it)|^4 dt - TP_4(\log T), \] where \(P_4(x)\) is a polynomial in \(x\) of degree four whose leading coefficient is \(1/(2\pi^2)\). The results \[ E_2(T) \ll T^{2/3}\log^8T, \int_0^TE_2^2(t) dt \ll T^2\log^{22}T, E_2(T) = \Omega_\pm(T^{1/2}) \] are the sharpest ones known. Each chapter is followed by Notes, where comments, clarifications, references etc. are given. The body of the text is followed by the bibliography, which contains 76 references. There is an author index and a subject index.

In conclusion, it should be said that this work is self-contained, written in a lucid and clear style, with many simplifications of previous proofs and results. It will be of great use both to novices and experts. This is a most welcome addition to the literature on analytic number theory, written by a renowned expert. The reviewer highly recommends this book to all interested readers.

The content of the book is as follows. Chapter 1. Non-Euclidean harmonics. This is the introductory chapter, laying down the groundwork. The author proves the spectral resolution of the non-Euclidean Laplacian \[ \Delta = -y^2((\partial/\partial x)^2 + (\partial/\partial y)^2) \] with minimum prerequisites. Poincaré, Eisenstein series and Maass wave forms are introduced and their basic properties are developed. Chapter 2. Trace formulas. Transformation formulas for the sums \[ \sum_{\ell=1}^\infty {1\over\ell}S(m,\pm n;\ell)\varphi\left({4\pi\over\ell} \sqrt{mn}\right) \] are proved, where \(m,n\) are arbitrary positive integers, \(\varphi\) is a suitable smooth function and \(S(m,n;\ell)\) is the Kloosterman sum. The results are expressed in terms of spectral sums of Fourier coefficients of holomorphic and real-analytic cusp forms over the full modular group \(\Gamma\). These trace formulas play an important rôle in subsequent investigations of \(\zeta(s)\).

Chapter 3. Automorphic \(L\)–functions. Hecke operators and Hecke series \(H_j(s)\) are discussed. The theory is later used to obtain results on bilinear forms of Hecke \(L\)–functions and a non-vanishing result on \(H_j(1/2)\). Spectral large sieve results are obtained, and the important upper bound \[ \sum_{\kappa_j\leq K}\alpha_jH_j^4(1/2) \ll K^2\log^{15}K \] is derived. Here as usual \( \{\lambda_j = \kappa_j^2 + {1\over4}\} \cup \{0\} \) is the discrete spectrum of the non-Euclidean Laplacian and \(\alpha_j = |\rho_j(1)|^2(\cosh\pi\kappa_j)^{-1}\), where \(\rho_j(1)\) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue \(\lambda_j\) to which the Hecke series \(H_j(s)\) is attached.

Chapter 4. An explicit formula. The last two chapters of the book are devoted to the investigation of \(\zeta(s)\). The main object of study is the power moment \[ {\mathcal Z}_k(g) = \int_{-\infty}^\infty|\zeta(1/2+it)|^{2k}g(t) dt, \] where \(k\) is a given natural number, \(g(r)\) takes real values on the real axis and there exists a large positive constant \(A\) such that \(g(r)\) is regular and \(\ll (|r|+1)^{-A}\) for \(|\text{ Im }r|\leq A\). The cases \(k = 1,2\) are treated in detail, of which \(k = 2\) (the fourth moment) is of special importance. The tour de force is Theorem 4.2, which gives the explicit spectral decomposition of \({\mathcal Z}_2(g)\), with no error terms. This decomposition reveals a structural view of a beautiful relation between \(\zeta(s)\) and the entire family of Hecke \(L\)–functions. Although it is possible to develop a theory that will treat the zeta- and allied functions simultaneously, this is not done, since a general argument would fail to exploit the peculiar properties of \(\zeta(s)\) and, on the other hand, it would involve considerable technicalities which might cloud the main ideas.

Chapter 5. Asymptotics. In the last chapter the author chooses the Gaussian weight \[ g(t) = (\sqrt{\pi}G)^{-1}\exp(-(T-t)G^{-2}) \] and evaluates \({\mathcal Z}_2(g)\) using Theorem 4.2. For \(T^{1/2}\log^{-D}T \leq G \leq T/\log T\) and an arbitrary constant \(D > 0\) one has \[ \begin{split} {1\over\sqrt{\pi}G}\int_{-\infty}^\infty |\zeta(1/2+it+iT)|^4\exp(-(t/G)^2) dt\\ = {\pi\over\sqrt{2T}}\sum_{j=1}^\infty\alpha_jH_j^3(1/2)\kappa_j^{-1/2} \sin\left(\kappa_j\log{\kappa_j\over4eT}\right)\exp\left(-{1\over4} (G\kappa_j/T)^2\right) + O(\log^{3D+9}T).\end{split}(1) \] The asymptotic formula (1) is used then to obtain results on the error term \[ E_2(T) = \int_0^T|\zeta(1/2+it)|^4 dt - TP_4(\log T), \] where \(P_4(x)\) is a polynomial in \(x\) of degree four whose leading coefficient is \(1/(2\pi^2)\). The results \[ E_2(T) \ll T^{2/3}\log^8T, \int_0^TE_2^2(t) dt \ll T^2\log^{22}T, E_2(T) = \Omega_\pm(T^{1/2}) \] are the sharpest ones known. Each chapter is followed by Notes, where comments, clarifications, references etc. are given. The body of the text is followed by the bibliography, which contains 76 references. There is an author index and a subject index.

In conclusion, it should be said that this work is self-contained, written in a lucid and clear style, with many simplifications of previous proofs and results. It will be of great use both to novices and experts. This is a most welcome addition to the literature on analytic number theory, written by a renowned expert. The reviewer highly recommends this book to all interested readers.

Reviewer: A.Ivić (Beograd)

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |