Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that .

Generalised prime systems with periodic integer counting function

We study generalised prime systems (both discrete and continuous) for which the `integer counting function' N(x) has the property that N(x) ¡ cx is periodic for some c > 0. We show that this is extremely rare. In particular, we show that the only such system for which N is continuous is the trivial system with N(x) ¡ cx constant, while if N has finitely many discontinuities per bounded interval, then N must be the counting function of the g-prime system containing the usual primes except for finitely many. Keywords and phrases: Generalised prime systems. I

'Quasi'-norm of an arithmetical convolution operator and the order of the Riemann zeta function

In this paper we study Dirichlet convolution with a given arithmetical function f as a linear mapping 'f that sends a sequence (an) to (bn) where bn = Pdjn f(d)an=d. We investigate when this is a bounded operator on l2 and ¯nd the operator norm. Of particular interest is the case f(n) = n¡® for its connection to the Riemann zeta function on the line 1, 'f is bounded with k'f k = ³(®). For the unbounded case, we show that 'f : M2 ! M2 where M2 is the subset of l2 of multiplicative sequences, for many f 2 M2. Consequently, we study the `quasi'-norm sup kak = T a 2M2 k'fak kak for large T, which measures the `size' of 'f on M2. For the f(n) = n¡® case, we show this quasi-norm has a striking resemblance to the conjectured maximal order of j³(® + iT )j for ® > 12 .

Typography papers, 9

This latest issue of the series of Typography papers opens with a beautifully illustrated article by the type designer Gerard Unger on ‘Romanesque’ letters. A further installment of Eric Kindel’s pathbreaking history of stencil letters is published in contributions by him, Fred Smeijers, and James Mosley. Maurice Göldner writes the first history of an early twentieth-century German typefounder, Brüder Butter. William Berkson and Peter Enneson recover the notion of ‘readability’ through a history of the collaboration between Matthew Luckiesh and the Linotype Company. Paul Luna discusses the role of pictures in dictionaries. Titus Nemeth describes a new form of Arabic type for metal composition. The whole gathering shows the remarkable variety and vitality of typography now.

Ovid, Remedia amoris 95: uerba dat omnis amor

Anagrams and syllabic wordplay of the kind championed by Frederick Ahl in his Metaformations have not always been favourably received by scholars of Latin poetry; I would hesitate to propose the following instance, were it not for the fact that its occurrence seems peculiarly apposite to the context in which it appears. That Roman poets were prepared to use such techniques to enhance the presentation of an argument by exemplifying its operation at a verbal level is demonstrated by the famous passage of Lucretius (DRN 1.907–14; also 1.891–2) in which the poet seeks to illustrate the tendency of semina … ardoris to create fire in wood by the literal presence of elements from the word for ‘fires’ (IGNes) in that denoting wood (lIGNum). A similar conception may underlie the association insinuated by the love elegists between amor and mors, suggesting that death is somehow ‘written into’ love: so Propertius declares laus in amore mori (2.1.47), while Tibullus appears to point to the lurking presence of death in the pursuit of love in the lines interea, dum fata sinunt, iungamus amores: | iam ueniet tenebris Mors adoperta caput (1.1.69–70) – so swift and unexpected is death's approach that it is already present in aMOReS in the preceding line. Ovid's awareness of the poetic potential of this kind of play (if that is the right word for it) is fully exhibited in his celebrated account of Echo and Narcissus in Metamorphoses 3, where the subject matter gives the poet ample scope to exploit the humorous and pathetic possibilities afforded by Echo's fragmented repetitions of the frustrated entreaties of her beloved.

An optimization problem concerning multiplicative functions

In this paper we study the problem of maximizing a quadratic form 〈Ax,x〉 subject to ‖x‖q=1, where A has matrix entries View the MathML source with i,j|k and q≥1. We investigate when the optimum is achieved at a ‘multiplicative’ point; i.e. where x1xmn=xmxn. This turns out to depend on both f and q, with a marked difference appearing as q varies between 1 and 2. We prove some partial results and conjecture that for f multiplicative such that 0

An example in Beurling's theory of generalised primes

In this paper we prove some connections between the growth of a function and its Mellin transform and apply these to study an explicit example in the theory of Beurling primes.

Gál-type GCD sums beyond the critical line

We prove that ∑k,ℓ=1N(nk,nℓ)2α(nknℓ)α≪N2−2α(logN)b(α) holds for arbitrary integers 1≤n1<⋯

Bergson leitor de lucrécio: as implicações existenciais do determinismo

Tomamos como objeto de análise a obra precoce de Bergson, os Extraits de Lucrèce, procurando mostrar que ao privilegiar as implicações existenciais negativas do determinismo, prefigura e justifica o fato de dedicar grande parte de seu pensamento filosófico posterior à crítica ao determinismo e à defesa da liberdade.

A natureza da alma e o clinamen: ação e liberdade em Lucrécio

The concept of freedom and his presupposition, the free will or libera voluntas in Lucretius relies on the notion of clinamen (declination), an occasional and random motion of atoms, inaccessible to experience. It is endowed with a complex movement provided by spontaneity, without the need for mechanical causes. The action of perceiving (sensus) is selfconsciousness, according to which this will, illuminated by previous experiences (sensory, intellectual or emotional) of the soul, takes advantage of the freedom and own spontaneity of atomic motions to drive them to a perceived direction and chosen. On the other hand, if we consider that the declination has a predominant role for the acts of will, we are facing other problems and questions. There is always the choice of a determined action and, therefore, even if the individual is facing a need to act, is possible choose to do not continue finish this action. Thereby, the will is related to conditions that are originate, ultimately, from the images formed randomly in space and impress the soul: the simulacra of desire and pleasure. The declination itself is a very important notion in this research, in order to emphasize the relationship between the freedom and the kinetic of elements. The approach developed in this work had as main objective to investigate the philosophy of nature and the soul in Lucretius, their constituents and movement, as well as demonstrating how the notion of clinamen articulates with the concepts of image, desire and pleasure, proposing a possible interpretation for the declination as an indeterminate and ethical foundation of freedom