Abstract: To make our exposition smooth, let us first introduce some notational conventions. For an algebraic number field K of finite degree, we denote by JK the set of all embeddings of K into C, and by IK the free Z-module generated by the elements of JK. We then put RIK = IK0 R and CIK = IK0 C. If p = EaPaU E IK with a E JK and PO E Z. we put xP = fl,(xa)Pfor 0 / x E K; this is meaningful for p e CIK if xa are all real and positive. Throughout the paper, we denote by D the complex upper half plane and by F a totally real algebraic number field of degree n. Now the zeta function to be studied in this paper, when suitably specialized, has the form

Abstract: For a totally real algebraic number field k, it is known that every (partial) zeta function of k is a finite sum of Dirichlet series which are regarded as natural generalizations of the Hurwits zeta function (see [1] and [2]). In this note we show that the similar result holds for arbitrary (not necessarily totally real) algebraic number field. At the time of the Bombay Colloquium (1979), H. M. Stark orally communicated to the author that he has obtained such a result for non-real cubic fields. His oral communication was an initial impetus to the present work. The author wishes to express his gratitude to Stark.

Abstract: To explain the main idea of this paper, and also to fix some notations, we start with reviewing the classical case of Riemann zeta function. As usual we set $$\varsigma \left( s \right) = \sum\limits_{n = 1}^\infty {{n^{ - s}}} \quad \left( {Re\,s >1} \right)$$ , $$\Gamma \left( s \right) = \int\limits_0^\infty {{x^{s - 1}}{e^{ - x}}dx\quad \left( {{\mathop{\rm Re} olimits} \,s >0} \right)}$$ .

Abstract: In this paper, the author presents a new approach to the subjects in the title, putting them in a new light. In fact, only integration by parts is used. This approach has two advantages: (1) it makes the p-adic theory seem even more natural, and (2) it is accessible to readers with only one year of basic calculus, making the subjects reachable in elementaxy courses. Many of the deepest properties of the Riemann zeta-function appear (with or without proof) in the work of Leonhard Euler. Euler's methods (which are summarized quite nicely in [1]) are extremely arithmetic in that they involve power series manipulations. As such, they are perfect for p-adic use (see N. Katz [4]). Euler's goal was to compute C(n) for n E Z (11. In this he succeeded in finding a closed form solution for all except the positive odd integers, a still open problem. It seems the first person to seriously consider c(s) for all positive reals was Dirichlet, though Euler certainly was in a position to do so had he chosen. It was Riemann who finally made the crucial step of considering ;(s), s complex, in order to describe ?(s) by its zeroes. Euler computed ?(-k) by using Abel summation in a very clever way; in fact he anticipated Abel by many years. This was first made rigorous by Riemann through complex methods. Here we present an elementary approach to these values that is rigorous by modem standards, yet could have been used by Euler as further justification for his computations. In doing so we obtain a simple means of analytic continuation for '(s) as well as a variety of integral formulas. In fact, we use only integration by parts. In light of this method, the p-adic theory becomes even more natural and seems almost forced. The author thanks Mike Rosen and Steve Galovich for their encouragement. We begin by recalling Euler's integral for n! (see [2]), n!= (-log x)" dx. Legendre set x = e-' to obtain n!= t"e-' dt; Received by the editors April 11, 1980 and, in revised form, August 27, 1980. 1980 Mathematics Subject Classification. Primary 1OH05; Secondary IOH08. ? 1981 American Mathematical Society 0002-9939/81/0000-0152/$02.25 513 This content downloaded from 157.55.39.147 on Wed, 21 Sep 2016 04:17:52 UTC All use subject to http://about.jstor.org/terms

Abstract: For orientation-reversing diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior. A diffeomorphism is Morse-Smale provided it is structurally stable and has a finite nonwandering set [P-S]. Several recent papers have explored the relationship between the topology of these maps and their dynamics. In [F] Franks showed that the periodic behavior of a Morse-Smale diffeomorphism was restricted by its homology zeta function. For the homotopy class of the identity on a compact surface Narasimhan proved that virtually any periodic behavior consistent with the homology zeta function does indeed occur [N]. For orientation-reversing maps there are obstructions other than the homology zeta function. Blanchard and Franks [B-F] have shown that if an orientation-reversing homeomorphism of S2 has periodic orbits which include two distinct odd periods, then the entropy of that map is positive. This implies that no orientationreversing Morse-Smale diffeomorphism on S2 can have distinct odd periods. The following theorem was conjectured in [B-F] and has been proven by Handel. THEOREM [H]. If f: M2 __ M2 is an orientation-reversing homeomorphism of a compact oriented surface of genus g, and iff has orbits with g + 2 distinct oddperiods, then the entropy off is positive. Thus an orientation-reversing Morse-Smale diffeomorphism on the torus has orbits with at most two different odd periods. In this paper we investigate whether there are any further restrictions on its periodic behavior. In [B] we showed that it suffices to consider the isotopy classes of the toral diffeomorphisms induced by (i 1) and (_1 ). We will show that there is a further obstruction in the former class but not in the latter. I would like to thank John Franks and Lynn Narasimhan for their contributions to this paper. Preliminaries. In this section we state our main result following some necessary definitions and background. We assume the reader is familiar with various standard terms and notation from dynamical systems. Further details are available in [B], [N] and [Ni]. Received by the editors November 26, 1979 and, in revised form, February 25, 1980. AMS (MOS) subject classifications (1970). Primary 58F09, 58F20. 'Research supported in part by Emory University Summer Research Grant. ( 1981 American Mathematical Society 0002-9947/81 /0000-0101 /$03.25

Abstract: We derive a series expansion for Ap(z) in which the terms of the expansion are simple rational functions of z. From a computational viewpoint, the new series is of interest in that it converges for all z not necessarily real valued, and is particularly rapid for values of z near the origin. From a mathematical viewpoint the series is of interest in that, although 4i(z) has poles at the negative integers and zero, the series is uniformly convergent in any finite interval a < Re(z) < b. A Series Expansion for A'(z). The derivative of the log gamma function, usually denoted by A/(z) (= d log r(z)/dz), is a regular function with simple poles at the negative integers and zero. No power series can therefore have infinite radius of convergence and this constraint limits the speed at which the coefficients in such a power series decrease. In this paper we give a series expansion for Ab(z) that has infinite radius of convergence despite the poles. The expansion is not a power series but converges uniformly to A'(z) in any finite interval. Convergence is particularly rapid for small z-say jzj < 1.0-and, in this range, ten terms of the expansion give accuracy to at least ten decimal places. For computational purposes, the recurrence relation 4'(z + 1) = A'(z) + 1/ z can be used for real z to obtain a value of z near the origin. Extremely large values of z are best dealt with by asymptotic expansions. The Taylor series expansion for 4p(1 + z) about z = 0 (Abramowitz and Stegun [1, Eq. 6.3.14]) is (1) 4P(1 + z) -'y + 2 (-1)r+ID(r + 1)zr, |zi < 1, r= 1 where y is Euler's constant, and t(r + 1) = Ew Ik-rI is Riemann's zeta function. The coefficients of the power series (1) do not tend to zero, so that the speed of convergence depends entirely on the magnitude of Izl. To obtain a more rapidly convergent series we rewrite (1) as a double series and regroup to obtain (2) ip(1 + z) = -y + I (-1)r+'Zr{Cr + ar/ (z + r)}, jzj < ?o, r= 1 where ar = r-r and Cr k= I r + Ik-r-l. It is readily shown that lim rr+ r = (e 1)1 0.582, so that both ar and cr diminish rapidly to zero. The series (2) therefore converges pointwise for all finite z, and it is therefore the analytic continuation of (1). To establish uniform convergence in any finite interval a < Re(z) < b, let gj(z) be the sum of the first n + 1 terms in (2) counting the constant, Received March 3, 1980. 1980 Mathematics Subject Classification. Primary 33A15; Secondary 65D20. i 1981 American Mathematical Society 0025-571 8/81/OOOQ-0020/$01.50 247 This content downloaded from 157.55.39.185 on Fri, 16 Sep 2016 05:35:55 UTC All use subject to http://about.jstor.org/terms

Abstract: Analysis on matrix groups and their homogeneous spaces is in a period analogous to that of Fourier, thanks to work of Harish-Chandra, Helgason, Langlands, Maass, Selberg, and many others Here we try to give a simple disccussion of Fourier analysis on the space Pn of positive n×n matrices, as well as on the Minkowski fundamental domain for Pn modulo the discrete group GL(n, Z) of integer matrice of determinant ±1 The main idea is to use the group invariance to see that the Plancherel or spectral measure in the Mellin inversion formula comes from the asymptotics and functional equations of the special functions which appear in the Mellin transform on Pn or Pn/GL(n, Z) as analogues of the power function ys in the ordinary Mellin transform For Pn, these functions are matrix argument generalizations of K-Bessel and spherical functions For Pn/GL(N,Z) these special functions are generalizations of Epstein zeta functions known as Eisenstein series

Abstract: Let M be a two-dimensional, compact manifold and g:Μ→ΜM be a diffeomorphism with a hyperbolic chain recurrent set. We find restrictions on the reduced zeta function p(t) of anyzero-dimensional basic set of g. If deg (p(t)) is odd, then p(1) = 0 (in ). Since there are infinitely many subshifts of finite type whose reduced zeta functions do not satisfy these restrictions, there are infinitely many subshifts which cannot be basic sets for any diffeomorphism of any surface.

Abstract: The root α of the equation can be determined by using the iterative formula provided that x 0is a good starting approximation to α and It is noted here that there are advantages in using the formula and, where λ is suitably chosen, this formula is equivalent to Aitken's; δ2extrapolation formula. The iterative technique is found to be successful in speeding up the convergence of alternating series and has also been applied to finding zeros of the Riemann Zeta Function.

Abstract: It is proved that the zeta function in m complex variables of a system of positive-definite forms of degree gd ≥ 2 with real coefficients extends meromorphically to the entire space ℂm.

Abstract: The thesis is divided into five sections: (a) Trigonometric sums involving prime numbers and applications, (b) Mean-values and Sign-changes of S(t)-- related to Riemann's Zeta function, (c) Mean-values of strongly additive arithmetical functions, (d) Combinatorial identities and sieves, (e) A Goldbach-type problem. Parts (b) and (c) are related by means of the techniques used but otherwise the sections are disjoint. (a) We consider the question of finding upper bounds for sums like ∑_PSN▒〖e(ap2)〗 and using a method of Vaughan, we get estimates which are much better than those obtained by Vinogradov. We then consider two applications of these, namely, the distribution of the sequence (αp2) modulo one. Of course we could have used the improved results to get improvements in estimates in various other problems involving p[superscript]2 but we do not do so. We also obtain an estimate for the sum ∑_PSN▒〖(ap3)〗 and get improved estimates by the same method. (b) Let N(T) denote the number of zeros of ς(s) - Riemann's Zeta function. It is well known that N(T) = L(T) + S(T), where L(T) = 1/2π Tlog(T/2π)-T⁄(2π+7⁄(8+0 ((1)⁄(T))))but the finer behaviour of S(T) is not known. It is known that S(t) ≪ log t ; ∫_o^t▒〖Slu)du〗 ≪ log t so that S(T) has many changes of sign. In 1942, A. Selberg showed that the number of sign changes of Set) for t ∈ (O,T) exceeds T (log T)1/3 exp(-A loglog T), (1) but stated to Professor Halberstam in 1979 that one can improve the constant 1/3 in (1) to 1 – ∈. It can be shown easily that the upper bound for the number of changes of sign is log T. We give a proof of Selberg's statement in (b), but in the process we do much more. Selberg showed that if k is a positive integer then ∫_T^(T+H)▒〖ls(t)l〖2k〗_dt 〗 = C CkH(loglog T) k ,{1+0( (loglogT)(-1)/2) } (2) where TT 1⁄2< H ≤ T[superscript]2 and C[subscript]k is some explicit constant in k. We have found a simple technique which gives (2) with the constant k replaced by any non-negative real number. Using this type of result, I prove Selberg's statement, with (log T)-∈ replaced by Exp (-A√loglogT (logloglogT) -□(1/2)). (c) I use the" method for finding mean-values above to answer similar questions for a class of strongly additive arithemetical functions. We say that f is strongly additive if (1) f(mn) = f(m) = f(n), if m and n are coprime, (2) f(p[superscript]a) = f(p) for all primes p and positive integer a. (d) This section contains joint work with Professor Halberstam and is still in its infancy. We have found a general identity and a type of convolution which serves to be the starting point of most investigations in Prime Number Theory involving the local and the global sieves. The term global refers to sieve methods of Brun, Selberg, Rosser and many more. The term local refers to things like Selberg's formula in the elementary proof of the prime number theorem, Vaughan's identity and so on. We have shown that both methods stem from the same source and so leads to a unified approach to such research. (e) I considered the question of solving the representation of an integer N in the form N = P_(1^2 )+ P _(2^2 ) +K P[subscript] 3, where the Pi’s are prime numbers. This problem was motivated by Goldbach's Problem and is exceedingly difficult. So I looked into getting partial answers. Let E(x) denote the numbers less than x not representable in the required form. Then there is a computable constant δ > 0 such that E(x) ≪ X i- δ To do this we use a method of Montgomery and Vaughan but the proof is long and technical, and we do not give it here. We show by sieve methods that the following result holds true: N = P_(1^2 ) + P _(2^2 ) +kP3P4P5. We have been unable to replace the product of three primes by two. Note: k is a constant depending on the residue class of N modulo 12.

Abstract: It is shown that non-trivial relations between certain values of the dilogarithm function can be obtained through the asymptotic comparison of coefficients of the expressions which appear in the Rogers-Ramanujan and Andrews-Gordon identities.

Abstract: and except for a simple pole at s = l , may be analy t ica l ly continued throughout the complex plane. The resemblance of ~(s,~) to Riemann's zeta-function, {(s), is in certain ways superf ic ial . For besides the two cases ~(s,I/2) = (2S-l)~(s) and ~(s, l) = ~(s), ~(s,~) possesses neither a functional equation nor an Euler product. I t is therefore not surprising that the zeros of these functions are distributed d i f ferent ly . For instance, we note the following:

Abstract: Recently, Solomon has introduced a zeta function which counts sublattices of a given lattice over an order ([5]). Let us recall the definition of this zeta function. Let I be a (finitedimensional) semisimple algebra over the rational fieldQ or over the j!?-adicfieldQp, and let A be an order in I. A is a Z-order when I is a Q-algebra, while A is a Zp-order when 2 is a Qp-algebra, where Zp is the ring of p-adic integers. Throughout this paper, p stands for a rational prime and the subscript p indicates the p-dAic completion. Let V be a finitelygenerated left I'-module, and let L be a full//-lattice in V. Solomon's zeta function is defined as

Abstract: In this paper we will consider the functions E(z, ρ) obtained by setting the complex variable s in the Eisenstein series E(z, s) equal to a zero of the Riemann zeta-function and will show that these functions satisfy a number of remarkable relations. Although many of these relations are consequences of more or less well known identities, the interpretation given here seems to be new and of some interest. In particular, looking at the functions E(z, ρ) leads naturally to the definition of a certain representation of SL2(R) whose spectrum is related to the set of zeroes of the zeta-function.