On Short Exponential Sums Involving Fourier Coefficients of Holomorphic Cusp Forms

By an exponential sum of the Fourier coefficients of a holomorphic cusp form we mean the sum which is formed by first taking the Fourier series of the said form,then cutting the beginning and the tail away and considering the remaining sum on the real axis. For simplicity’s sake, typically the coefficients are normalized. However, this isn’t so important as the normalization can be done and removed simply by using partial summation. We improve the approximate functional equation for the exponential sums of the Fourier coefficients of the holomorphic cusp forms by giving an explicit upper bound for the error term appearing in the equation. The approximate functional equation is originally due to Jutila [9] and a crucial tool for transforming sums into shorter sums. This transformation changes the point of the real axis on which the sum is to be considered. We also improve known upper bounds for the size estimates of the exponential sums. For very short sums we do not obtain any better estimates than the very easy estimate obtained by multiplying the upper bound estimate for a Fourier coefficient (they are bounded by the divisor function as Deligne [2] showed) by the number of coefficients. This estimate is extremely rough as no possible cancellation is taken into account. However, with small sums, it is unclear whether there happens any remarkable amounts of cancellation.

Tekijännimet Mikael Agricolan teosten kielessä. Henkilötarkoitteisten johdosten merkitykset, funktiot ja rakenteet

My research deals with agent nouns in the language of the works of Mikael Agricola (ca. 1510–1557). The main tasks addressed in my thesis have been to describe individual agent noun types, to provide a comprehensive picture of the category of agent nouns and to clarify the relations between different types of agent nouns. My research material consists of all the agent nouns referring to persons in the language of Agricola’s works, together with their context. The language studied is for the most part translated language. Agent nouns play an important role both in the vocabulary of natural language and in broader sentence structures, since in a text it is constantly necessary to refer to actors re-ferring to persons in the text. As a concept and a phenomenon, the agent noun is widely known in languages. It is a word formed with a certain derivational affixes, which typical-ly refers to a person. In my research the agent noun category includes both deverbal and denominal derivatives referring to persons, e.g. kirjoittaa > kirjoittaja (to write > writer), asua > asuva (to inhabit > inhabitant), imeä > imeväinen (to suck > suckling), juopua > juopunut (to drink > drunkard), pelätä > pelkuri (to fear > one who fears ‘a coward’), apu > apulainen (help/to help > helper); lammas > lampuri (sheep > shepherd). Besides original Finnish expressions, agent noun derivatives taken as such from foreign languages form a word group of central importance for the research (e.g. nikkari, porvari, ryöväri, based on the German/Swedish for carpenter, burgher, robber). Especially important for the formation of agent nouns in Finnish are the models offered by foreign languages. The starting point for my work is predominantly semantic, as both the criteria for collecting the material and the categorisation underlying the analysis of the material are based on semantic criteria. When examining derivatives, aspects relating to structure are also inevitably of central importance, as form and meaning are closely associated with each other in this type of vocabulary. The alliance of structure and meaning can be described in an illustrative manner with the help of structural schemata. The examination of agent nouns comprises on the one hand analysis of syntactic elements and on the other, study of cultural words in their most typical form. The latter aspect offers a research object in which language and the extralinguistic world, referents, their designations and cultural-historical reality are in concrete terms one and the same. Thus both the agent noun types that follow the word formation principles of the Finn-ish language and those of foreign origin borrowed as a whole into Finnish illustrate very well how an expression of a certain origin and formed according to a certain structural model is inseparably bound up with the background of its referent and in general with semantic factors. This becomes evident both on the level of the connection between cer-tain linguistic features and text genre and in relation to cultural words referring to per-sons. For example, the model for the designations of God based on agent nouns goes back thousands of years and is still closely linked in 16th century literature with certain text genres. This brings out the link between the linguistic feature and the genre in a very con-crete manner. A good example of the connection between language and the extralinguistic world is provided by the cultural vocabulary referring to persons. Originally Finnish agent noun derivatives are associated with an agrarian society, while the vocabulary relat-ing to mediaeval urbanisation, the Hansa trade and specialisation by trade or profession is borrowed and originates in its entirety from vocabulary that was originally German.

Amerikkalaiset folkloristiikan alkujäljillä

Timothy Lloyd (American Folklore Society) vierailulla Suomessa.

Kasveja ja kasvinnimityksiä 1500-luvulta

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