this paper we will first prove the theorem above, then extend it to a form from which one can find as many terms of the expansion as desired, and finally discuss the relationship of our work to earlier results of Moser and Wyman [3]. * Research supported by the United States Office of Naval Research. 2. Proof of theorem 1 We will write i n (s) for P n j=1 j \Gammas , which, if s ? 1, is the nth partial sum of the Dirichlet series for the Riemann zeta function. The following computation will be the basis for the developments in the sequel. We have, from the familiar generating function, = (1 + x)(1 + ) \Delta \Delta \Delta (1 + log(1 + )g sj exp f(\Gamma1) g: (1) In the last member we wish to replace all of the i n 's by their asymptotic expansions in powers of n and log n. These expansions are (e.g., [5], p. 124) i n (s) log n + fl + d(1; j)n \Gammaj ; if s = 1; i(s) + (1 \Gamma s) \Gammas+1 j=1 d(s; j)n \Gammas\Gammaj+1 ; if s ? 1, (2) where d(s; j) = \Gamma B j s + j \Gamma 2 and the B j 's are the Bernoulli numbers. By combining (1) and (2), we now have exp x(log n + fl + d(1; j) \Theta i(s) + 1 (1 \Gamma s)n d(s; j) s+j \Gamma1 (3) in the sense that for each fixed k, the coefficient of x on the left has, for its complete asymptotic series as n !1, the coefficient of x on the right. Now we can prove theorem 1. To do that we need only keep the portion of the development in powers of n and log n that does not approach zero on the right side of (3), along with the size of the first neglected term. If we take the coefficient of x on both sides of (3) we obtain x(log n+fl) \Psi\Psi The second exponential factor above is ...

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ABSTRACT. Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Bombieri–Lagarias, Ma´slanka, Coffey, Báez-Duarte, Voros and others. We apply the theory of Nörlund-Rice integrals in conjunction with the saddlepoint method and derive precise asymptotic estimates. The method extends to Dirichlet Lfunctions and our estimates appear to be partly related to earlier investigations surrounding Li’s criterion for the Riemann hypothesis.

We investigate some algorithms that produce Bernoulli, Euler and Genocchi polynomials. We also give closed formulas for Bernoulli, Euler and Genocchi polynomials in terms of weighted Stirling numbers of the second kind, which are extensions of known formulas for Bernoulli, Euler and Genocchi numbers involving Stirling numbers of the second kind. 1

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We present a character-free proof of the divisible code bound and some applications. 1 Introduction A linear code is called divisible if the weights of its codewords have a common divisor larger than 1. When the divisor \Delta is relatively prime to the alphabet size, the code is equivalent to a \Delta-fold replicated code, with perhaps some additional 0-coordinates [14, Theorem 1]. Thus if the length of the code is n, the dimension of the code is at most n=\Delta. The divisible code bound encompasses the circumstance that the characteristic of the alphabet divides \Delta. Let the alphabet for codes be GF (q), where q is a power of the prime p. For an integer x, let the p-adic valuation v(x) be the exponent of the highest power of p that divides x, taking v(0) = 1: The divisible code bound states that if C is a linear code over GF (q) whose nonzero codeword weights are among the m multiples (b \Gamma m+ 1)\Delta; : : : ; b\Delta of the divisor \Delta, then the dimension k of C satisf...

The camera placement problem concerns the placement of a fixed number of point-cameras on the integer lattice of d-tuples of integers in order to maximize their visibility. We give a caracterization of optimal configurations of size s less than 5 d and use it to compute in time O(s log s) an optimal abstract configuration under the assumption that the visibility of a configuration is computable in constant time. 1980 Mathematics Subject Classification: 68U05, 52A43 CR Categories: F.2.2, I.3.5 Key Words and Phrases: Art gallery problems, Camera placement problem, Density, Exchange method, Integer lattice, Integer optimization, Prime number, Visibility. y Centrum voor Wiskunde en Informatica, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands and Carleton University, School of Computer Science, Ottawa, ONT, K1S 5B6, Canada. Research supported by NSERC grant #907002. (kranakis@scs.carleton.ca) Laboratoire d'informatique de l'Ecole normale sup'erieure, ura 1327, Cnrs, 45 rue d'Ulm,...

For the creation operator a † and the annihilation operator a of a harmonic oscillator, we consider Weyl ordering expression of (a † a) n and obtain a new symmetric expression of Weyl ordering w.r.t. a † a ≡ N and aa † = N + 1 where N is the number operator. Moreover, we interpret intertwining formulas of various orderings in view of the difference theory. Then we find that the noncommutative parameter corresponds to the increment of the difference operator w.r.t. variable N. Therefore, quantum (noncommutative) calculations of harmonic oscillators are done by classical (commutative) ones of the number operator by using the difference theory. As a by-product, nontrivial relations including the Stirling number of the first kind are also obtained. 1

Abstract. Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindelöf, which belong to an attractive but largely forgotten chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotics of certain finite difference sequences come out as a byproduct of our approach.

this paper we consider two problems on the generalized Bernoulli polynomials B