Abstract: Here, arrow (i) refers to our previous papers [9, 12] and [13], in which we have shown that a fixed primitive cusp form f of Sk(F I(N)) has congruences with other cusp forms of Sk(F~(N)) modulo the special value at s = k of a certain zeta function off . In this paper, we will construct the second arrow for the cusp form f associated with a Hecke character of an imaginary quadratic field K, and consequently obtain the third. Namely, in w167 6 we will show, as in the works of Coates-Wiles [3] and Robert [23], that the split primes of K which divide this special value are irregular in an appropriate sense. In this case, the zeta function o f f mentioned above is a Hecke L-function of K. Such a criterion of irregularity for Hurwitz numbers has already been obtained in [3] and [233. However, our result together with those of [3] and [23] does not cover all the L-values of Hecke L-functions of K whose algebraicity was given by Damerell [5] (for details, see below). Our method is analogous to that of Ribet [21] (see also Wiles [35]), who used congruences between cusp forms and Eisenstein series to prove the non-vanishing of certain eigenspaces (relative to the action of Gal(Q/Q)) of the p-primary part of the class group of the field of p-th roots of unity. By adopting this approach, we will obtain such an information of eigenspaces again in our case (see Theorem 0.1 below). To be more specific, let 2 be a Hecke character of K satisfying

Abstract: Denote by N ( t ) the number of zeros β + iγ of ζ( s ), s = σ + it , with 0 t . It is well known that where

Abstract: © Foundation Compositio Mathematica, 1982, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: For a population with an unknown number M of equally likely classes, an estimator for M from random samples is given. The results are applied to the problem of determining the total amount of coinage in past civilizations.

Abstract: Poisson's summation formula tells us that the Fourier transform of the sum of Dirac's (J-functions supported by the integral points is again the sum of Dirac's (^-functions supported by the integral points. In this paper we first consider the converse problem, that is, we characterize a distribution which is a sum of distributions supported by the integral points and whose Fourier transform is again of the same form. Using this result we give another proof of the classical theorem of Hamburger on the characterization of the zata function of Riemann. We also show a generalization of the result to the zeta function associated to the imaginary quadratic field Q(^ — 1) -. In what follows, we use the notation 2 (f) or f to denote the Fourier transform of f normalized in the form I f (x) exp(27Tv — 1 ) dx. We denote by Z>£ (resp., xa} (a= (al9 •--, am) eZJ) the differential operator d™/da%* — da%? (resp., the monomial x"l~>x°tf), where Z+ m denotes the set of ra-tuple of non-negative integers and \a\ = XI #/• j=i We denote Dirac's d-f unction supported at x — n by 8(x — 7i) and its derivative Dax8(x-n) by 8(a}(x-n). Now the first result is stated as follows:

Abstract: Zero-free regions of thekth derivative of the Riemann zeta function ζ(k)(s) are investigated. It is proved that fork≥3, ζ(k)(s) has no zero in the region Res≥(1·1358826...)k+2. This result is an improvement upon the hitherto known zero-free region Res≥(7/4)k+2 on the right of the imaginary axis. The known zero-free region on the left of the imaginary axis is also improved by proving that ζ k)(s) may have at the most a finite number of non-real zeros on the left of the imaginary axis which are confined to a semicircle of finite radiusr k centred at the origin.

Abstract: We determine when a polynomial is the reduced zeta function of a basic set of a Smale diffeomorphism of a compact surface.

Abstract: On Shimura's correspondence for modular forms of half-integral weight.- Period integrals of cohomology classes which are represented by Eisenstein series.- Wave front sets of representations of Lie groups.- On p-adic representations associated with ?p-extensions.- Dirichlet series for the group GL(n).- Crystalline cohomology, Dieudonne modules and Jacobi sums.- Estimates of coefficients of modular forms and generalized modular relations.- A remark on zeta functions of algebraic number fields.- Derivatives of L-series at s = 0.- Eisenstein series and the Riemann zeta function.- Eisenstein series and the Selberg trace formula I.

Abstract: One gives a new proof to the Leopoldt-Kubota-Iwasawa theorem regarding the possibility of the p-adic interpolation of the values of the Riemann zeta-function and of the Dirichlet L-functions at negative integral points. To this end, for each root ɛ ≠ 1 of unity one introduces and one investigates the numbers Cn(ɛ) which arise in the expansion $$\frac{{\varepsilon - 1}}{{\varepsilon e^z - 1}} = \sum\limits_{n = 0}^\infty {\frac{{C_n (\varepsilon )}}{{n!}}Z^n }$$ One proves a generalization of the Kummer congruences for the Bernoulli numbers.

Abstract: By Rankin's method one constructs a meromorphic continuation and a functional equation for the convolution of two Dirichlet series corresponding to cusp forms of weight O. One investigates the summator function of the coefficients of this convolution.

Abstract: We investigate the arithmetic character of the Fourier coefficients of a Hilbert-Siegel modular form which is an eigenfunction of all Hecke operators. We prove the analytic continuability and a functional equation for the corresponding zeta function.

Abstract: Trace identities are derived for the operator p2+xq, q positive even. For q=4 these identities can be used to estimate the ground state energy within an accura cy of about 10−4.

Abstract: We describe extensive computations which show that Riemann's zeta function t(s) has exactly 200,000,001 zeros of the form a + it in the region 0 < t < 81,702,130.19; all these zeros are simple and lie on the line a = . (This extends a similar result for the first 81,000,001 zeros, established by Brent in Math. Comp., v. 33, 1979, pp. 1361-1372.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser's rule" are given.

Abstract: The classical mean value theorem for Dirichlet's polynomials states thatsee H. L. Montgomery [7]. This formula is very useful in the theory of the Riemann zeta-function ζ(s). From the approximate functional equationwhere | χ(½ + it)| = 1, u, v ≥ 1, 2πuv = t (see E. C. Titchmarsh [8]) it follows that χ(½ + it) can be well approximated by Dirichlet's polynomials of length N< t½.