99b:11097 11M06 (11Y35) Paris, R. B.(4-ABDU-DM); Cang, S.(4-ABDU-DM) An asymptotic representation for $\zeta(\frac 12+it)$. (English. English summary) Methods Appl. Anal. 4 (1997), no. 4, 449--470. Generalizing a classical method of Riemann, the authors derive an approximate formula for Riemann's zeta-function. The zeta-function is first expressed by a version of the approximate functional equation due to A. F. Lavrik [Trudy Moskov. Mat. Obsc. 18 (1968), 91--104; MR 38 #4424]. It is actually an identity involving values of the incomplete gamma-function, and if these are given by asymptotic expansions in terms of the complementary error function, the result becomes a formula of the Riemann-Siegel type. However, the truncation of the zeta series at the usual cut-off value $[(t/2\pi )\sp {1/2}]$ is done "smoothly" by a weight function dropping rapidly to zero in a neighbourhood of this point. The (very good) accuracy of the basic formula is demonstrated by a number of numerical examples and tables. Reviewed by Matti Jutila _________________________________________________________________ 98k:11118 11M26 (11N05) Coppola, G.(I-SLRNE-II); Laporta, M. B. S.(I-NAPL-AM) A new "weighted" form of the Riemann-von Mangoldt explicit formula. (English. English summary) Note Mat. 14 (1994), no. 2, 263--275 (1997). The Riemann-von Mangoldt explicit formula relates the number of primes up to $x$ with a sum over the zeros $\rho =\beta +i\gamma$ of the Riemann zeta-function. For applications one often truncates the sum over the zeros to include only terms with $\vert \gamma\vert \le T$, which introduces an error term in the formula. One can also weight the sum over zeros in various ways. With minor changes one can introduce characters and obtain explicit formulas involving zeros of Dirichlet $L$-functions. Let $\Lambda(n)$ be the von Mangoldt function, which is $\log p$ if $n=p\sp m$ and zero otherwise. Then with $\chi(n)$ denoting a Dirichlet character, let $\psi(x,\chi) = \sum\sb {n\le x} \Lambda(n)\chi(n)$, and take $$\psi\sb 0(x,\chi) = {1\over 2}( \psi(x+0,\chi)+\psi(x-0,\chi)).$$ The truncated Riemann-von Mangoldt formula is $$\multline\psi\sb 0(x,\chi)=\\\epsilon(\chi)x - \sum\sb {\vert \gamma\vert \le T} w\bigg({\vert \gamma\vert \over T}\bigg){x\sp \rho\over \rho}-\epsilon\sb 1(\chi)\log x -\epsilon\sb 2(\chi) +R(x,T,\chi),\endmultline$$ where the sum is over zeros of $L(s,\chi)$, $\chi$ is a primitive character, $\epsilon(\chi)$ is $1$ if $\chi=\chi\sb 0$ is the principal character and is zero otherwise, $\epsilon\sb 1(\chi) =1$ if $\chi$ is not principal and $\chi(-1)=1$ and is zero otherwise, $$\epsilon\sb 2(\chi)=\lim\sb {s\to0}\bigg({L'\over L}(s,\chi) - {\epsilon\sb 1(\chi) \over s}\bigg),$$ $w$ is a weight, and $R$ is an error term. In the classical case $w\equiv 1$, and for $2\le T\le x$ and $1\le q \le x$ the error term satisfies $R\ll(x\log\sp 2x)/T$. In the current paper the following result is obtained. Let $Y$ be a positive integer, and let $\Phi\sb Y(\tau)$ be any $C\sp Y$ function on $[T/2,T]$ which vanishes along with its first $Y-1$ derivatives at $\tau = T/2$ and $\tau =T$, and satisfies the growth condition $$ \left( \int\sb {T/2}\sp T\Phi\sb y(\tau) \,d\tau \right)\sp {-1} {d\sp r\Phi\sb Y(\tau) \over d\tau\sp r } \ll\sb Y {1\over T\sp {r+1} } \quad {\rm for} 0\le r \le Y.$$ Given $\Phi\sb Y$, we now define the weight $$w\sb Y(u) = \cases 1 &{\rm if} 0\le u\le\frac 12,\\ \int\sb u\sp 1\Phi\sb Y(Tv)\,dv / \int\sb {1/2}\sp 1 \Phi\sb Y(Tv) \,dv & {\rm if} \frac 12\le u \le 1. \endcases$$ Then for $16\le N\le x\le 2N$, $4\le T\le N/4$, $1\le M \le T/4$, we have $R(x,T, \chi) = R\sb 1 + R\sb 2$ where $$\multline R\sb 1 =\\{1\over \pi} \sum\sb { x - {MN\over T} < n\le x +{MN\over T} } \Lambda(n) \chi(n) {\rm sgn }(x-n) {2\over T} \int\sb {T/2}\sp T\int\sb {\tau \vert \log {x\over n}\vert }\sp \infty {\sin u\over u} \,du \,d\tau ,\endmultline $$ and $R\sb 2$ is an error term which is smaller than the classical estimate for $R(x,T,\chi)$ and decreases as we make the parameter $M$ get larger. In the special case of $q=1$, $\chi = \chi\sb 0$, and $M=1$, $$R\sb 2 \ll\sb Y{ N\log N \over T \log {N\over T}}. $$ This result generalizes recent work of Kaczorowski and Perelli where the weight $w\sb Y(u)$ was obtained from $\Phi\sb Y(\tau) \equiv 1$. The authors intend to use their results to study primes in short intervals. Reviewed by Daniel A. Goldston _________________________________________________________________ 98b:11131 11Z05 (11M06 81Q50) Bhaduri, R. K.(3-MMAS-PA); Khare, Avinash(6-IOP); Reimann, S. M.(DK-CPNH-OL); Tomusiak, E. L.(3-SK-EP) The Riemann zeta function and the inverted harmonic oscillator. (English. English summary) Ann. Physics 254 (1997), no. 1, 25--40. [ORIGINAL ARTICLE] The authors point out some exotic property features of the Riemann zeta function and the inverted harmonic oscillator. First, they observe that the phase of the Riemann zeta function shifts discontinually every time that $\zeta(z)$ changes sign. This shift can be Lorentz smoothed by considering an increase of a parameter $\sigma>\frac12$. $\sigma=\frac12$ corresponds to the critical line for the Riemann zeta function. Increasing $\sigma$, the jumps in the phase fade gradually, as is easily expected. In the sequel the authors study a two-dimensional saddle harmonic oscillator model with potential $V(x,y)=(m/2)(\omega\sb x\sp 2x\sp 2-\omega\sb y\sp 2y\sp 2)$, a model with highly unstable orbits. It is analysed by means of the Gutzwiller trace formula [M. C. Gutzwiller, Chaos in classical and quantum mechanics, Springer, New York, 1990; MR 91m:58099], and it is found that a Lorentzian smoothing of the density of states exhibits the same features as in the case of the Riemann zeta function. Whereas the study of the quantum mechanical saddle oscillator by means of periodic orbit theory is an interesting investigation, the similarity of Lorentzian smoothing in the density of states, respectively in the density of zeros for $\zeta(z)$, seems to be coincidental. Reviewed by Christian Grosche _________________________________________________________________ 98a:11121 11N05 (11M26) Stechkin, S. B.; Popov, A. Yu.(RS-MOSC) Asymptotic distribution of prime numbers in the mean. (Russian) Uspekhi Mat. Nauk 51 (1996), no. 6(312), 21--88. Define $R(x)$ by $\sum\sb {n\leq x}\Lambda(n)=x+R(x)$, where $\Lambda$ is the von Mangoldt function. Let $R\sp +(x)=\max\{0,R(x)\}$ and $R\sp -(x)=-\min\{0,R(x)\}$. It is well known that $R(x)=\Omega\sb ±(x\sp {1/2}\log\log\log x)$, $\int\sp x\sb 1R(u)du=\Omega\sb ±(x\sp {3/2})$ without any unsettled condition, and that $R(x)\ll x\sp {1/2}(\log x)\sp 2$, $\int\sp x\sb 1\vert R(u)\vert dux\sp {3/2}/200$ and $\int\sp {Ax}\sb xR\sp ±(u)du>x\sp {3/2}$, where $A$ is some constant. First the problems are transformed into those of zeros of the Riemann zeta-function via the explicit formula. Then ingenious methods in Fourier analysis are used with the aid of some known results on zeta zeros. The authors seem to be unaware of J. Pintz's work [in Elementary and analytic theory of numbers (Warsaw, 1982), 411--417, PWN, Warsaw, 1985; MR 87m:11085], in which some similar results were established by means of the power-sum method. Reviewed by Hiroshi Mikawa _________________________________________________________________ 97j:35114 35P20 (11M06 28A80 58G25) He, Christina Q.(1-CAR); Lapidus, Michel L.(1-CAR) Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function. (English. English summary) Mem. Amer. Math. Soc. 127 (1997), no. 608, x+97 pp. This paper continues Lapidus' detailed study of the Dirichlet eigenvalue problem (DE) $\Delta u=-\lambda u$ on a bounded domain $\Omega\subset\bold R\sp n$, where $u=0$ on $\partial\Omega$, which may be fractal. The aim is to understand the asymptotics of the eigenvalue counting function $N(\lambda)=\#\{\lambda\sb i\leq\lambda\colon\ \lambda\sb i$ solves (DE)$\}$ for this problem. Weyl's classical theorem gives the first term in the asymptotic expansion of $N(\lambda)$ with the constant determined by the volume of $\Omega$. Recent work has shown that if the boundary has Minkowski dimension $D$ and finite upper Minkowski content then the second term is of order $\lambda\sp {D/2}$. In general there is a sharp remainder only in the case $n=1$ when there is a connection with the zeros of the Riemann zeta function. In this paper these results are extended to domains with boundaries which do not necessarily have the usual Minkowski content. As in the construction of a more general Hausdorff measure function the idea of Minkowski content can be generalised to functions other than just powers. In the case $n=1$ this allows sharp remainder estimates for $N(\lambda)$ to be expressed in terms of the gauge function for the Minkowski content and the situation where there are sharp remainder estimates is characterised. For $n\geq 2$ the order of the remainder estimate is obtained when the upper generalised Minkowski content is finite. Examples are given to illustrate the results. The results of this paper were announced earlier by the authors [Math. Res. Lett. 3 (1996), no. 1, 31--40; MR 97i:58176]. Reviewed by Ben Hambly _________________________________________________________________ 97e:11095 11M06 (11Y60) Choudhury, Bejoy K.(1-LMS) The Riemann zeta-function and its derivatives. (English. English summary) Proc. Roy. Soc. London Ser. A 450 (1995), no. 1940, 477--499. Summary: "Formulas for higher derivatives of the Riemann zeta-function are developed from Ramanujan's theory of the `constant' of series. By using the Euler-Maclaurin summation methods, formulas for $\zeta\sp {(n)}(s)$, $\zeta\sp {(n)}(1-s)$ and $\zeta\sp {(n)}(0)$ are obtained. Additional formulas involving the Stieltjes constants are also derived. An analytical expression for the error bounds is given in each case. The formulas permit accurate derivative evaluation and the error bounds are shown to be realistic. A table of $\zeta'(s)$ is presented to 20 significant figures for $s=-20(0.1)20$. For rational arguments, $\zeta(1/k),\zeta'(1/k)$ are given for $k=-10(1)10$. The first ten zeros of $\zeta'(s)$ are also tabulated. Because the Stieltjes constants appear in many formulas, the constants were evaluated freshly for this work. Formulas for the $\gamma\sb n$ are derived with new error bounds, and a tabulation of the constants is given from $n=0$ to 100." _________________________________________________________________ 96m:11070 11M06 (11-02) Laurin\v cikas, Antanas(LI-VILN) Limit theorems for the Riemann zeta-function. (English. English summary) Mathematics and its Applications, 352. Kluwer Academic Publishers Group, Dordrecht, 1996. xiv+297 pp. $149.00. ISBN 0-7923-3824-3 Several books on the Riemann zeta-function have appeared in recent years, but the present one seems to be the first devoted solely to its probabilistic aspects. Starting from the basis of standard university courses, this monograph provides an excellent introduction to topics like the value distribution of the zeta-function or its "universality" properties. To a considerable extent, the material covered represents the author's own research in probabilistic number theory. Illuminating additional information may be found in the notes after each chapter, and there is an extensive bibliography. The necessary background from probability theory is given in Chapter 1. Then the main objects of study, the Dirichlet series and polynomials, are introduced in Chapter 2. Basic properties of Riemann's zeta-function and Dirichlet's $L$-functions are surveyed; it may be a bit misleading for a non-specialist that the zero-free region for the zeta-function (in Theorem 8.6) is not given in the sharpest known form. The actual topic of the book begins in Chapter 3 with a discussion of limit theorems for the modulus of the zeta-function in the half-plane $\sigma \geq \frac 12$ (in the first place near the critical line). The theory of moments of the zeta-function (going back to Ramachandra and Heath-Brown) plays an important role here. In the next chapter, the value distribution of the zeta-function is studied more generally. Chapter 5 deals with limit theorems of the zeta-function in the space of analytic functions, as a preparation for a proof of Voronin's universality theorem in Chapter 6. Then, in Chapter 7, a limit theorem for the zeta-function in the space of continuous functions is established under the assumption of the Riemann hypothesis. In the last two chapters, preceding limit theorems pertaining to the zeta-function are generalized to wider classes of functions, first to Dirichlet $L$-functions, and finally to Dirichlet series with multiplicative coefficients. Reviewed by Matti Jutila Cited in reviews: 98c:11086 ________________________________________________________________ 96f:11110 11M36 (11F72 11M41 30B50) Jorgenson, Jay(1-YALE); Lang, Serge(1-YALE); Goldfeld, Dorian(1-CLMB) Explicit formulas. Lecture Notes in Mathematics, 1593. Springer-Verlag, Berlin, 1994. viii+154 pp. $30.00. ISBN 3-540-58673-3 The work under review contains two separate parts, the first (longer) part authored by Jorgenson and Lang, and the second part by Goldfeld. The work of Jorgenson and Lang is a very general approach to the investigation of explicit formulas. An explicit formula is viewed as establishing the relation between the sum of a suitable test function over the prime powers and the sum of the Mellin transform of the function over the zeros of the zeta function, plus an analytical functional evaluated on the test function. The authors then establish such explicit formulas for a general class of zeta functions, which are assumed to have an Euler sum, and a functional equation whose fudge factors are of regularized product type. The precise definition of this fundamental class is given in the first section of Chapter II. The results obtained in this fashion apply to many of the known examples of explicit formulas, which arise, so to speak, in more natural settings. This general viewpoint was in fact initiated by the authors, and therefore the current work should be consulted in conjunction with their previous as well as forthcoming works, which are listed in the bibliography. Also, attention should be focused on the comprehensive and quite detailed introduction given by the authors themselves, where some of the philosophy is expounded. A brief chapter-by-chapter summary of the work follows. Chapter I gives a number of asymptotic estimates of general regularized harmonic series. This chapter is preparatory in nature relative to the general framework. However, as the authors point out, much of the hard analysis is contained therein, paving the road for later developments of the main subject, which often assumes a formal appearance. Chapter II, after the definition of the fundamental class mentioned above, is devoted to a general Cramer-type theorem (as an explicit formula). The explicit formulas are then established, under certain growth conditions on Fourier transforms, in Chapter III. For a precise statement see, in particular, Theorem 2.1. The next two chapters are dual to each other---Chapter IV gives a general version of the theta inversion formula for members of the fundamental class (and hence satisfying a functional equation), while Chapter V shows that a theta series satisfying an inversion formula gives rise to a Dirichlet series satisfying an additive functional equation. Finally, Chapter VI presents a generalization of Fujii's theorem, which follows the author's generalized Cramer theorem. It is worth noting that, while many of the topics in the work under review have already been considered in the earlier works of the authors, they are not merely repeated or summarized here. The tendency is that they are considered in still more general settings in this work. Also, a number of applications of the results to more classical settings are given throughout. In the contribution by Goldfeld, he observes, first of all, that the classical theory of automorphic forms (over ${Q}$) may be viewed as based upon the spectral properties of the group ${\rm GL}(2,{\bf Q})$ acting on ${\rm GL}(2,{A})$, where ${A}$ is, as usual, the adele ring of ${Q}$. Consider the space of $L\sp 2$ functions on ${\rm GL}(2,{Q})\backslash {\rm GL}(2,{A})$. It is well known, for instance, that there is the orthogonal decomposition of this space into cusp forms, Eisenstein series, and residues of Eisenstein series. The author views the group ${\rm GL}(2,{Q})$ as generated by additions and one inversion. He then develops an analogous theory of automorphic forms for a group defined over ${Q}$ by multiplications and one inversion, which acts on the idele group ${A}\sp \times$. The above-mentioned orthogonal decomposition of the $L\sp 2$ space exists in this case, and in fact the analogy goes much further. Each of the basic functions (cusp forms, etc.) permits a Mellin expansion giving rise to a canonical zeta function, and there is a generalization of the Rankin-Selberg convolution and the Gelbart-Jacquet lift. The trace formula for this space is exactly the explicit formula of A. Weil, which explains the title of this work. While the discrete spectrum of the Laplacian on the space ${\scr L}\sp 2({\rm GL}(2,{Q})\backslash {\rm GL}(2,{A}))$ is given by the zeros of the Selberg zeta function, the same object for the author's $L\sp 2(\Upsilon)$ is given by the zeros of the Riemann zeta function. The construction of all of the above-mentioned objects corresponding to the classical ones is given in the paper. The reader interested up to this point is now invited to consult the original work, for example, for an equivalent statement of the Riemann hypothesis in this new setting. Reviewed by Ze-Li Dou Cited in reviews: 97f:11072 96m:11075 ________________________________________________________________ 96b:11120 11M26 Farmer, David W.(1-MSRI) Counting distinct zeros of the Riemann zeta-function. (English. English summary) Electron. J. Combin. 2 (1995), Research Paper 1, approx. 5 pp. (electonic). Let $\xi(s)$ be the Riemann $\xi$-function, $N(T)$ be the number of zeros of $\xi(\sigma+it)$ in the region $0<\sigma<1$, $0kN(T)$, with $k=0.63952\cdots$. Furthermore, given the bounds on $\beta\sb j$, this result is best possible. The author gives two methods for determining the lower bounds for $N\sb {\rm d}(T)$. The first method is based on the inequality $$N\sb {\rm d}(T)\geq\sum\sp R\sb {r=1}\bigg[\frac{M\sb {\leq r}(T)}{r(r+1)}+\frac{M\sb {\leq R+1}(T)}{R+1}\bigg],$$ which was obtained by J. B. Conrey, A. Ghosh and S. M. Gonek [in Number theory, trace formulas and discrete groups (Oslo, 1987), 185--199, Academic Press, Boston, MA, 1989; MR 90h:11077]. The second method eliminates the loss inherent in the first method, and uses the inequality $$N\sp {(n)}\sb {\rm s}(T)\leq\sum\sp {n+1}\sb {j=1}M\sb j(T)+n\sum\sp \infty\sb {j=n+2}M\sb j(T)/j.$$ Reviewed by Yasushi Matsuoka _________________________________________________________________ 96a:11146 11Y35 (11E45 11M26 11N05) Odlyzko, Andrew M.(1-BELL) Analytic computations in number theory. (English. English summary) Mathematics of Computation 1943--1993: a half-century of computational mathematics (Vancouver, BC, 1993), 451--463, Proc. Sympos. Appl. Math., 48, Amer. Math. Soc., Providence, RI, 1994. This is a valuable survey of analytic computations in number theory which have been carried out since the discovery by Riemann of the close connection between the distribution of the prime numbers and complex zeros of the (Riemann zeta) function $\zeta(s)=\sum\sb {n=1}\sp \infty n\sp {-s}$. First, the Riemann hypothesis (stating that all the complex zeros of $\zeta(s)$ have real part $\frac 12$), and algorithms and computations for its verification in a finite part of the so-called critical strip, are discussed. Some other questions are listed which go beyond the Riemann hypothesis, like the distribution of the primes, and the size of gaps between consecutive primes: they depend on the vertical distribution of the complex zeros of $\zeta(s)$. The Riemann hypothesis can also be formulated and verified for related functions like Dirichlet $L$-functions (important for the study of the distribution of primes in arithmetic progressions), and the Epstein zeta function (important for the study of quadratic forms and number fields). Zeros of $\zeta(s)$ have been used in disproofs of various conjectures, like the conjecture that $\pi(x)$, the number of primes $\le x$, is bounded above by the function li$(x)=\int\sb 0\sp x du/\log u$, for all $x\ge2$, and in the derivation of explicit bounds of number-theoretic functions like $\pi(x)$. The paper closes with a discussion of an analytic method for computing exact values of $\pi(x)$ for large $x$, and a combinatorial method which theoretically requires more time ($x\sp {2/3+o(1)}$) than the analytic method ($x\sp {1/2+o(1)}$), but is simpler to implement. The combinatorial method was used to compute $\pi(10\sp {18})=24\,739\,954\,287\,740\,860$. Reviewed by Herman J. J. te Riele _________________________________________________________________ 95a:11081 11M41 (11Y35) Spira, Robert Some zeros of the Titchmarsh counterexample. (English. English summary) Math. Comp. 63 (1994), no. 208, 747--748. Titchmarsh gave an example of a function which satisfies the exact same functional equation as that of a Dirichlet $L$-function, viz. $$f(s)\coloneq 5\sp {{1/ 2}-s}2(2\pi)\sp {s-1}\Gamma(1-s)\cos(\tfrac1 2s\pi)f(1-s).$$For this function, however, he could show that it must have zeros with $\sigma>1$ and, from a theorem of Voronin, also zeros in the critical strip but off the critical line. The example function involves the real number $\theta\coloneq {1\over 2}\tan\sp {-1}(2\cos(2\pi/5))$, and is defined as $$f(s)\coloneq \sum\sb {k=0}\sp {\infty}(5k+1)\sp {-s}-(5k+4)\sp {-s}+ \tan\theta\sum\sb {k=0}\sp {\infty}(5k+2)\sp {-s}-(5k+3)\sp {-s}.$$ As any zeros of the Riemann zeta function off the critical line would have to have very large imaginary part, it is reasonable to wonder whether this counterexample has accessible zeros. The author shows that it does. There are zeros at $0.808517+85.699348 i$, $0.650860+114.163343 i$, $0.574356+166.479306 i$ and $0.724258+176.702461 i$. These values are reported to have been checked in various ways, and the reviewer can add to this that Euler-MacLaurin based calculation of $f$ at the first of these zeros agreed to seven places. The paper reports that no zeros with $\sigma>1$ and $t<200$ were found, but remarks that Titchmarsh's proof had to assume $t$ large. From the evidence so far, it seems reasonable to hope that these zeros of $f$ with $\sigma>1$ may not be utterly beyond reach of calculation. Reviewed by Douglas Hensley _________________________________________________________________ 94a:11138 11M41 (11M06) Ramachandra, K.(6-TIFR-SM) On Riemann zeta-function and allied questions. Journées Arithmétiques, 1991 (Geneva). Astérisque No. 209 (1992), 57--72. The author begins with three conjectures. The first of these asks that $\int\sp T\sb 0\vert \sum\sb {n\leq N}a\sb nn\sp {it}\vert \sp 2dt\gg T\vert a\sb 1\vert \sp 2$ as soon as $T\geq T\sb 0$. The second conjecture is a stronger version of the first. The third conjecture, which we shall not state in full here, suggests conditions under which a generalized Dirichlet series $\sum\sp \infty\sb 1a\sb n\lambda\sp {-s}\sb n$ has $\gg T\log T$ zeros in a region $\sigma\geq\frac 12-\delta$, $\vert t\vert \leq T$. The bulk of the paper is then devoted to a discussion of results that establish special cases or weaker versions of the conjectures. In consequence there are, for example, estimates for mean values of the Riemann zeta-function. Proofs of the results, which are mainly contained in joint work with Balasubramanian, are not given. Reviewed by D. R. Heath-Brown Cited in reviews: 95j:11081 _________________________________________________________________ 93h:11096 11M06 (11-02 11L15 11M26 11N05) Karatsuba, A. A.(RS-AOS-NT); Voronin, S. M.(RS-AOS-NT) The Riemann zeta-function. Translated from the Russian by Neal Koblitz. de Gruyter Expositions in Mathematics, 5. Walter de Gruyter & Co., Berlin, 1992. xii+396 pp. $112.00. ISBN 3-11-013170-6 The aims of this book are twofold: first, to serve as an introduction to the theory of the Riemann zeta-function with its number-theoretic applications, and second, to acquaint readers with certain advances of the theory not covered by previous comprehensive treatises of the zeta-function such as the classic of E. C. Titchmarsh [The theory of the Riemann zeta-function, second edition, Oxford Univ. Press, New York, 1986; MR 88c:11049], or a more recent monograph of A. Ivic [The Riemann zeta-function, Wiley, New York, 1985; MR 87d:11062]. Thus the choice of the more advanced material is intentionally selective; for instance, there is no discussion of mean value problems and results for the zeta-function. The headings of the chapters give an idea of the contents: I. The definition and simplest properties of the Riemann zeta-function, II. The Riemann zeta-function as a generating function in number theory, III. Approximate functional equations, IV. Vinogradov's method in the theory of the Riemann zeta-function, V. Density theorems, VI. Zeros of the zeta-function on the critical line, VII. Distribution of nonzero values of the Riemann zeta-function, VIII. $\Omega$-theorems. In addition, there is an extensive appendix containing various auxiliary results, and a bibliography of 172 references. Most of the new or less standard material, mainly originating from the research of the authors, can be found in the last four chapters. To give a few examples, results of Selberg type on zeros lying on or near the critical line are given as "local" versions; the distribution of the zeros of the Davenport-Heilbronn function, the Hurwitz zeta-function and zeta-functions of quadratic forms (all having a functional equation but not an Euler product) is discussed in detail; further, there are theorems about the "universality" of the zeta-function and allied functions, as well as about the independence of $L$-functions; and finally, in the last chapter, a multidimensional $\Omega$-theorem is proved in addition to more standard results of this kind. These examples also indicate that the scope of this well-written book is by no means restricted to the Riemann zeta-function. It spans the range successfully from elementary theory to topics of recent and current research. Reviewed by Matti Jutila Cited in reviews: 97d:11001 96g:11112 _________________________________________________________________ 93c:11067 11M06 Karatsuba, A. A. On the zeros of Riemann's zeta-function on the critical line. International Symposium in Memory of Hua Loo Keng, Vol. I (Beijing, 1988), 117--162, Springer, Berlin, 1991. This paper is concerned with two different problems in analytic number theory, and discusses results obtained by the author and others in the last 20 years. The first problem concerns the integral $$\theta=\int\sp \infty\sb {-\infty}\cdots\int\sp \infty\sb {-\infty}\Big\vert \int\sp 1\sb 0e\sp {2\pi i(\alpha\sb nx\sp n+\cdots+\alpha\sb 1x)}\,dx\Big\vert \sp {2k}\,d\alpha\sb 1\cdots d\alpha\sb n,$$ which is related to a problem of solving in integers the systems of equations $(x\sb 1)\sp j+\cdots+(x\sb k)\sp j=(y\sb 1)\sp j+\cdots+(y\sb k)\sp j$, $1\leq j\leq n$. Hua showed $\theta$ converges for $k>(n/4)(n+2)$ and proposed the problem of finding the best result. In 1978 Arkhipov, Chubarikov, and the author solved this problem by showing that $\theta$ converges for $k>(n/4)(n+1)+\frac12$ and diverges for smaller $k$. A proof of this result is given in this paper, and an application to partial differential equations is proved. The second problem considered in this paper is concerned with detecting zeros of the Riemann zeta function $\zeta(s)$, $s=\sigma+it$, on the critical line in short intervals. Let $N(T)$ denote the number of zeros of $\zeta(s)$ in the strip $0<\sigma<1$, $00$, one has $N\sb 0(T+H)-N\sb 0(T)\geq c(N(T+H)-N(T))$ with $H=T\sp a$, $\frac12\leq a\leq 1$, thus proving that a positive proportion of the zeros are on the critical line. Selberg also stated: "The results of the present paper do not pretend to be the best which can be obtained by these and similar methods. In fact, several things suggest that the condition $a>\frac12\cdots$, if we use still more sophisticated arguments, may be replaced by $a>\vartheta$ where $\vartheta<\frac12$." In 1984 the author proved this result [Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 3, 569--584; MR 85i:11070] with $a=\frac{27}{82}=\frac13-\frac1{246}$. In the paper under review he discusses the complicated calculations used to obtain this result (which is referred to repeatedly as Selberg's conjecture). Some "almost all" results are also proved with smaller $a$. Reviewed by Daniel A. Goldston _________________________________________________________________ 91m:11067 11M26 Fujii, Akio(J-RIK) An additive theory of the zeros of the Riemann zeta function. Proc. Japan Acad. Ser. A Math. Sci. 66 (1990), no. 5, 105--108. Let $\rho=\beta+i\gamma$ denote a zero of the Riemann zeta-function in the critical strip. This paper announces a number of results concerning sums over pairs of zeros. The first result is that $$\sum\Sb 0<\gamma,\;\gamma'\leq T\\\gamma+\gamma'\leq T\endSb 1=\frac{1}{8\pi\sp 2}T\sp 2\log\sp 2\,T+c\sb 1T\sp 2\log T+c\sb 2T\sp 2+O \bigg(\frac{T\log\sp 2 T}{\log\log T}\bigg),$$ where $c\sb 1$ and $c\sb 2$ are explicit constants. Letting $N(T)=\sum\sb {0<\gamma\leq T}1$ as usual, the above sum may be written as $\sum\sb {0<\gamma\leq T}N(T-\gamma)$. The Riemann-von Mangoldt formula provides a very accurate formula for $N(T)$, and therefore we can evaluate this last sum by partial summation, which leads to the result. Assuming the Riemann hypothesis, the error term can be reduced to $O(T\log T)$. The author next obtains, subject to the Riemann hypothesis, asymptotic formulas in certain ranges for the more complicated sums $$\sum\Sb 0<\gamma,\gamma'\leq T\\\gamma+\gamma'\leq T\endSb x\sp {i(\gamma+\gamma')}\quad{\rm and}\;\sum\Sb 0<\gamma,\gamma'\leq T\\ \gamma+\gamma'\leq T\endSb e\left({b\over 2\pi}(\gamma+\gamma')\log\left({\gamma+\gamma'\over 2\pi e\alpha}\right)\right),$$ where $e(u)=e\sp {2\pi iu}$ and $\alpha>0$. An unconditional result is also obtained for the first sum with the summand replaced by $x\sp {\rho+\rho'}$. In each case the result can be obtained by partial summation using the known formula for the appropriate expression summed over $\gamma$. Detailed proofs and generalizations will appear elsewhere. Reviewed by Daniel A. Goldston _________________________________________________________________ 89j:11083 11M06 (11M26 11Y35 65E05 68Q25) Odlyzko, A. M.(1-BELL); Schönhage, A.(D-TBNG) Fast algorithms for multiple evaluations of the Riemann zeta function. Trans. Amer. Math. Soc. 309 (1988), no. 2, 797--809. The authors give a new method of evaluating the zeta function $\zeta(\sigma +it)$, with time and storage both $O(T\sp {(1/2)+\varepsilon})$ for $\sigma$ fixed and $T\leq t\leq T+T\sp {1/2}$. The method consists in evaluating Dirichlet series, and several of their derivatives, at a set of regularly spaced values; this is done by using the fast Fourier transform to reduce the problem to the evaluation of rational functions. The method is applicable to other number-theoretic functions, such as $\pi(x)$, but then further analytical work, to give good error bounds, and suitable contours of integration, etc., is a prerequisite. It is stated that the method would enable the testing of the Riemann hypothesis for the first $n$ zeros to be done in $O(n\sp {1+\varepsilon})$ operations, as opposed to $O(n\sp {3/2})$ by earlier methods, provided that no multiple zeros, or too closely spaced zeros, of $\zeta(s)$ occur. Reviewed by H. J. Godwin _________________________________________________________________ 87e:11102 11M26 (11-04 11Y35 30C15) van de Lune, J.(NL-MATH); te Riele, H. J. J.(NL-MATH); Winter, D. T.(NL-MATH) On the zeros of the Riemann zeta function in the critical strip. IV. Math. Comp. 46 (1986), no. 174, 667--681. From the summary: "Very extensive computations are reported which extend and, partly, check previous computations concerning the location of the complex zeros of the Riemann zeta function. The results imply the truth of the Riemann hypothesis for the first 1 500 000 001 zeros in the critical strip. All these zeros are simple. Various tables are given with statistical data concerning the numbers and first occurrences of Gram blocks of various types; the numbers of Gram intervals containing $m$ zeros, for $m=0,1,2,3$ and $4$; and the numbers of exceptions to `Rosser's rule' of various types. Graphs of the function $Z(t)$ are given near five rarely occurring exceptions to Rosser's rule, near the first Gram block of length 9, near the closest observed pair of zeros of the Riemann zeta function, and near the largest (positive and negative) found values of $Z(t)$ at Gram points." Part III has been reviewed [MR 85e:11062]. Reviewed by Jurgen G. Hinz © Copyright American Mathematical Society 1999