Manuel Sartori

Manuela E B Giolfo

Philippe Cassuto

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

"Studies in Semitic Language and Linguistics" 88)

INTRODUCTION

Sagittarius luminum de thesauris linguae et de clari sermonis mari, 1 one can not fail to notice the important place that Pierre Larcher occupies within our sciences, and especially with reference to Arabic Grammar and Arabic Linguistics. Author of approximately 200 items in this field (not to mention his output in the literary domain), he has been described as the alter ego of Raḍī al-Dīn al-ʾAstarābāḏī (d. 688/1289) 2 of whom he is, without any doubt, the best interpret. Thus it appears more than natural that leading scholars and distinguished young researchers come together to pay homage to Pierre Larcher for a life devoted to the enlightening exploration of the 'ocean' of Arabic and Semitic Linguistics in the broadest and deepest sense ever reached.

What strikes one about Pierre is the fact that his research interests are extraordinarily wide-ranging and that at the same time he shows an extremely high level of specialist skill in handling with them: from pre-Islamic poetry to linguistics, from morphology to logic, from lexicography to semiology, from Indo-European to Semitistic, from epigraphy to history of language, from theology to philology, from philosophy to politilical theory, from Qurʾānic studies to sociolinguistics, from fuṣḥā to ʿāmmiyya, and from 'Arab linguistics' to 'Arabic linguistics'. Even more striking is the fact that Pierre was able to isolate some issues of pivotal interest within such an immense universe, notably that of pragmatics, and that of 'Arab metalanguages'. It seems that nothing relevant could ever escape his keen sight and sharp intellect. One sees this for example in his work on the relationship between linguistics and the other sciences in the Arab-Islamic society, and also in his treatment of the concepts of semantic 'coordination' and semantic 'subordination', in his deep reflections on moyen arabe and arabe moyen, and also on 'enunciation and interaction' in Oswald Ducrot's theory, as well as on the grammatical works of a rhetorician such as ʿAbd al-Qāhir al-Ǧurǧānī (d. 471/1078).

Last but not least, Pierre is not only an accomplished scholar, in the fields of both linguistics and poetry; he is also a great man of culture, in every sense and acceptance of this word. He is an incomparable scientific partner, responsive and open to ideas that complement his, and at the same time vigorous in defending his own views. For some of us, Pierre is not only a model but also a catalyst for ideas in the positive sense, and more, he is a creator, allowing the creativity of colleagues with whom he works. In sum, he is an example to follow.

1

We are more than indebted to our colleagues and friends Claude Gilliot and Francesco Zappa for their kind support in composing the Latin title -starting from the editors' idea of Rāmī al-lumaʿ fī ʿulūm al-luġa -which perfectly suits the scientific dignity of Pierre Larcher.

2

As Antoine Lonnet writes in the review he made for Bulletin de la Société de Linguistique de Paris (39/2, 1994, 355) of the special issue of Bulletin d'Études Orientales, 'De la grammaire de l'arabe aux grammaires des arabes ' (B.E.O., 43, 1991) edited by Pierre Larcher.

The first section of the present dedicated volume, Semitic Linguistics, proposes a journey From Sem to Qedar, from Biblical Hebrew to Semitic and Afroasiatic passing by Arabic and Early Aramaic.

Philippe Cassuto, in 'Few Switching of Labials in Biblical Hebrew', takes further some ideas from his article 'The expression of peace in Hebrew' in Redefining Peace in the Twenty-first Century (Nancy, 2001). Pierre Larcher had written in the same vein an article entitled 'The expression of peace in Arabic.' However, Pierre considered two such expressions in Arabic: Silm and Salam, whereas in Hebrew Philippe Cassuto noted only the traditional Shalom. In the present chapter, Cassuto argues that Shalom and Shalwa had very similar meanings, and the contrast between them is purely lexical. It consists in switching the letters mem and waw, two labial letters. In considering the question, it appears that many of these permutations exist in Semitic languages; and the object and focus of Cassuto's contribution is about this phenomenon: it is possible to trace the etymology of the Jewish month Marheshwan by comparing with the Akkadian. Another example is in Ugaritic. In the cuneiform tablets, the deity Shpsh is mentioned. By swapping p with the labial m, it is easy to identify Shpsh with the Hebrew Shemesh, sun, equivalent of the Arabic word šams. This article proposes to consider other examples and, if possible, to establish whether or not rules for the labial permutations in Semitic languages can be defined.

Joseph Dichy argues in 'The Analytics of Writing, Exemplified by Arabic, the Youngest of Semitic Scripts' that the categorization of Semitic writings as alphabetic, consonantal or even syllabic still remains problematic. The alternative "analytic paradigm" considers that writing systems result from metalinguistic abilities at work in the linguistic community. It involves two interwoven aspects: (a) analyzing language, according to pleremic and cenemic conventions and (b) representing the result through a visual artefact (semiographic characterization). The basic secondary convention is that of the graphic word-form, a complex unit related, in the semiography of Arabic writing, to variation in the shape of letters, fundamentally divided into final vs. non-final.

In 'Arabian faḫr and mubālaġa of High Rhetorical Value: a New Comprehensive View of Nemara Inscription' Manfred Kropp recalls that the inscription of Nemara has been, ever since its discovery in 1902, object of scientific research in more than 100 contributions. Part of a cenotaph dedicated to the Arab king Marʾ al-Qays Ibn ʿAmr, dated 328 AD, the text of five lines engraved in a tabula ansata on a lintel of basalt is written in late (Classical) Nabataean script but exhibiting transition to early Arabic script. The language, though, is clearly Arabic, even if not identical with later Classical Arabic. The document is a primary source for events in the Arabian Peninsula and its adjacent regions in the fourth century AD as well as for the history of the Arabic script and language. Thus the study of it concentrated mainly on elucidating the precious historical and linguistic details. What has been neglected to a certain extent is the general character of the text and its evident function for the contemporary reader. The article proposes to interpret it as an early and already accomplished masterpiece of Arabian faḫr and mubālaġa of high rhetorical value. From this the author derives two guidelines for reading the passage. The function requires a fluent and well constructed syntactical texture concentrated without deviations on the deeds and exploits of this "accomplished" Arab king. As for the historical facts they have to be taken under the cautious premises of being reported in the context of Arabian mubālaġa.

In 'Dia-planar diffusion: Reconstructing early Aramaic-Arabic Language Contact', Jonathan Owens gives three explanations for relatedness between historical linguistic stages: inheritance, diffusion or independent parallel development. In his paper he argues that in the period leading up to and entering the early Islamic era, Aramaic played an important role in influencing different variants of Arabic. By focusing on the phonological domain, he adduces data from all varieties of Arabic, as well as -on the Aramaic side -from Old and Middle Aramaic eras.

Next, Lutz Edzard presents in 'The masʾala zunbūriyya in a Semitic and Afroasiatic Perspective' the comparative Semitic and Afroasiatic scenario on the one hand and the typological comparison with Germanic and Romance on the other. As for the initial masʾala, the solution offered by al-Kisāʾī clearly must be admitted at least as a valid alternative. Sībawayhi was bound by the Baṣran tradition, which would not tolerate ʾiyyā-in predicate position, and simply could not allow for a variety of possibilities, in the way that al-Kisāʾī could. In sum, the masʾala zunbūriyya continues to be relevant for modern linguistic theory, and vice versa.

The second section of the present volume, Arabic Grammatical Tradition, is a Travel along Grammar and its Representations, and is dedicated to matters of morpho-syntactic, syntactic, semantic, pragmatic, rhetoric and logic nature, the treatment of which reveals how 'grammar', and the place it occupies within the system of the sciences of language, is related to 'models of representation', i.e. the philosophical and hermeneutical presuppositions underlying any analyses.

In an article following the theme developed by L. Edzard on case ending, Jean-Patrick Guillaume's '« Man Zaydan ? » À propos de quelques cas curieux de ḥikāya chez Sībawayhi' argues that the concept of ḥikāya ("literal quotation") plays a minor role in the Arabic grammatical theory, usually in order to explain an anomaly in the distribution of case markers. In many cases, the data treated in this context are inherited from Sībawayhi's Kitāb, and reflect a living, spontaneous oral usage characteristic of Old Arabic, a fact which became increasingly difficult for later grammarians to grasp as Old Arabic evolved into Classical Arabic, a purely scholarly language. Guillaume's paper focuses on two different sets of facts mentioned by Sībawayhi, in which ḥikāya functions in a dialogue in order to point a violation of the relevance principle, usually with a nuance of 'disapprobation' (ʾinkār). It aims to show that Sībawayhi's analysis, in spite of the rather primitive metalanguage he uses, is quite self-consistent and shows a remarkable perceptiveness of the pragmatic dimension of language.

In 'Inflectional Ending by Means of Short Vowels Among Arab Grammarians: Clues for the Deconstruction of a Grammatical Ideology', Manuel Sartori aims to give his own argument, based on some Arab grammarians, to show how the sacrosanct desinential inflection (ʾiʿrāb) would actually be a grammar construct, a creed which grammarians, whether believers or convinced disciples of believers, invite us to take as an untouchable reality. Dealing only with inflectional endings by means of short vowels, Sartori's paper recalls the strength of the pause (waqf) compared to effective realization of the inflection and focuses its analysis on the rules of case ending agreement between the noun and the attributive qualifying adjective. It shows then that the Ancient grammarians, whether Zaǧǧāǧī (d. 337/949), Fārisī (d. 377/987), Ibn Ǧinnī (d. 392/1002), or ʿAbd al-Qāhir al-Ǧūrǧānī (d. 471/1078), when addressing an agreement that was already well-known and highly regulated, demonstrate other possibilities, the number and variety of which show how little relevance the inflectional endings have, thus providing arguments for those sceptical about ʾiʿrāb. Ramzi Baalbaki's 'One Word, two Functions. The Concept of Functional Replacement in Traditional Syntactic Analysis' starts with the observation that, according to the Arabic rules of parsing, each noun, and by extension each nominal or verbal sentence which fulfils a grammatical function that can be expressed in the form of a noun (e.g. mufrad), is assigned one, and only one, maḥall. Thus the notion of maḥall represents a specific grammatical function performed by one element of the construction. However, Baalbaki notes there are constructions in which one noun apparently performs two functions and thus occupies two maḥalls. In the frequently cited construction ḍarbī l-ʿabda musīʾan, musīʾan is a circumstantial accusative (ḥāl) which also fulfils the function of the predicate (ḫabar), hence the expression ḥāl saddat masadd al-ḫabar. Functional replacement, expressed by terms such as sadda masadd, nāba manāb, ʾaġnā, etc., has its roots in Sībawayhi's Kitāb. Baalbaki's paper discusses Sībawayhi's use of this analytical tool, traces its expansion in the Arabic grammatical tradition, and identifies thirteen types of constructions which grammarians interpret based on it. In line with their interest in standardization and rule formulation, later authors in particular introduce complex rules pertaining to some of these types. Constructions that are extremely unlikely to be used in actual speech were also made up in order to examine the theoretical implications of functional replacement. The paper also argues that the main purpose of the grammarians in introducing the notion of functional replacement is to defend the theory of 'one-element-one-maḥall' since the admission that one element can have two maḥalls would shatter one of their most essential axioms in syntactical analysis. Francesco Binaghi's 'Ẓarf and mafʿūl fī-hi: Really Two of a Kind? Some Notes on Zaǧǧāǧī's Treatment' investigates the categories of ẓarf and mafʿūl fī-hi as they occur in the Arabic grammatical tradition. The analysis of some passages and examples from Ibn al-Sarrāǧ's ʾUṣūl and Zaǧǧāǧī's Ǧumal leads to conclude that these two terms are not synonyms: ẓarf indicates the semantic role of locative and temporal, whereas mafʿūl fī-hi designates a syntactic function and thus represents only a subset of all the possible occurrences of ẓarf. Ibn al-Sarrāǧ defines the mafʿūl fī-hi as the verbal adjunct of time and place in the accusative case, whereas Zaǧǧāǧī defines it in terms of the scope it has (the predicative core of the sentence). Zaǧǧāǧī also makes use of these two terms for the definition of the noun as a part of speech, especially in his ʾĪḍāḥ. At this level of linguistic analysis, the two terms reverse their extension: in Zaǧǧāǧī's theoretical structuration of the different occurrences that define a noun, the ẓarf becomes a subset of mafʿūl fī-hi since the mafʿūl (in the broader sense) constitutes one of the possible occurrences of the noun.

The travel continues with Nadia Anghelescu's 'The Role of the Metaphor in the Interpretation of Prepositions: the Arabic min and the French de'. Her chapter contains an interpretation of the manner in which specialists in the science of language, old Arab grammarians and modern French researchers, have analyzed prepositions with similar functions: Arab min and French de. Anghelescu is concerned with the way in which these grammarians approach grammaticalization within a category of linguistic elements which already fulfil a certain grammatical function, i.e. prepositions. Prepositions with a similar initial meaning (beginning in space and then in time) acquire new meanings, of a similar type in both languages, among which the indefinite quantifier function takes an important place. All these evolutions result from the conceptualization of space. Research itself resorts to the spatial metaphor, and the ingenious interpretation of the preposition min delivered by a 13 th century author, Raḍī al-Dīn al-ʾAstarābāḏī, is conclusive in that respect.

Next, Catherine Pinon's 'Une corrélation retrouvée : nécessaire vs possible' directly echoes a major chapter of Pierre Larcher's Le système verbal de l'arabe classique: « Une corrélation oubliée : nécessaire vs possible ». By focusing on the expression of 'possibility' and 'necessity' within a digital corpus of contemporary written Arabic, the paper shows that the perfect kāna is used to mark 'necessity', whist the imperfect, in its three forms, is used to mark sometimes 'necessity' and sometimes 'possibility'. It is argued that utterances can be modalized to different degrees and that modalization can be lexically or structurally reinforced. As an example, qad yakūnu and rubbamā yakūnu structures are examined in detail and hypotheses are proposed about the nuances in expressing the modality of 'possibility'. The paper observes also the presence, in contemporary written Arabic, of apocopated verbal forms yakun used with a modal value of 'necessity' in utterances that would previously resort to the perfect form. As for negations, it is noted that lā yakūnu appears as strongly modalizing vis-à-vis laysa, and that the negation lān yakūna sometimes carries a modal value in addition to the temporal value of 'future'. Although fundamental, the issue of 'modalities' is rarely present in grammars. Pinon's study comes complete with a review of the issue in Arab and Arabic grammars, which she compares with the results of her analysis. Manuela E.B. Giolfo and Wilfrid Hodges' 'The System of the Sciences of the Arabic Language by Sakkākī: Logic as a Complement of Rhetoric' finds its inspiration in Pierre Larcher's chapter 'Arabic Linguistic Tradition II: Pragmatics', in J. Owens (ed.) The Oxford Handbook of Arabic Linguistics (2013). In Sakkākī's (d. 626/1229) Miftāḥ al-ʿulūm "The key to the sciences", the sciences of language are presented as a complex system whose core includes the two sciences of morphology (ṣarf) and of syntax (naḥw), that is to say grammar; the two sciences of meanings (maʿānī) and of expression (bayān) -that is to say rhetoric -, and the two sciences of definition (ḥadd) and of argumentation (istidlāl) -that is to say logic. The complexity of the system lies in the fact that syntax (ʿilm al-naḥw) finds its complement (tamām) in semantics (ʿilm al-maʿānī) which in turns finds its complement in logic (ʿilmā al-ḥadd wa-l-istidlāl). An axis 'syntaxsemantics-logic' is thus drawn which brings logic within the field of linguistics. The 'systemic' intersection between rhetoric and grammar, and the 'meta-systemic' intersections between rhetoric and literature from one part and that between rhetoric and religious sciences from the other have been a subject of strong interest. However, the same cannot be said for another intersection, 'systemic' by Sakkākī: that between rhetoric, namely semantics (ʿilm al-maʿānī) and logic (ʿilmā al-ḥadd wa-l-istidlāl). This latter is what Giolfo and Hodges explore in their paper, mainly basing themselves on Sakkākī's Miftāḥ, on Rāzī's (d. 606/1209) logic (Mulaḫḫaṣ), as well as on Ibn Sīnā's (d. 428/1037) logical works.

Finally, this section ends with a grammatical discussion between ʾAbū al-ʿAlāʾ al-Maʿarrī (d. 449/1058) and some angels in Martino Diez's 'Teaching Arabic to the Angels: a Scherzo by al-Maʿarrī on Heavenly Morphology'. In his paper, Diez translates and comments on a section of the introduction to the Risālat al-Malāʾika where the poet, together with some men of letters, tries to convince the Guardian of Paradise to grant them access to heavenly joys because of their linguistic skills. He discusses in particular the derivation and the morphological behaviour of some objects that can be found in Heaven, such as the Tūbā tree or the houris. This introduction, a deeply ironic text, targeting Islamic popular beliefs concerning the Afterworld, grammar and grammarians and, most importantly, the author himself, offers an interesting picture of the method followed in ʿilm al-taṣrīf ('morphology'), a field of scholarship which received considerable attention by the first generations of grammarians, but came later to be partially neglected in favour of other disciplines.

As a follow-up to the previous contribution, the third section of this dedicated volume, Arabic and Semitic Lexicology could have been entitled In a Garden paved with Words. In this Garden, three topics in relation with lexical issues are discussed: first etymology, then the issue of synonymy or polysemy, and finally the question of Arabic as a medium for other languages as well as its writing in Latin characters. Georgine Ayoub first, with 'L'emprunt dans le dictionnaire arabe des premiers siècles', examines the treatment of loanwords in some theoretical works of the first three centuries AH. Between anthropological considerations (that of the relation to the other) and religious ones (that of the sacrilegious relation to the sacred text), this study shows how two complementary approaches are drawn at the end of the 8 th century, the first proposing to detect by a phonological characterization the neologism (muḥdaṯ, muwallad) whether foreign or forged, the second to describe the linguistic process at work in "Arabization" understood as the linguistic integration of a foreign word. Ayoub shows how lexical study is part of the cultural context in religious sciences of the time to explain the vocabulary of the Qurʾān and studies in three dictionaries the analysis of twelve terms the first exegetes said foreign. Weipert conducts a semantic study that reveals that more than two-thirds of ʾufʿūlatun pattern words belong to a single semantic field related to speech acts and human forms of verbal expression. He then shows that the last third is in fact composed by quite a few old genuine Arabic words of unknown etymology, some foreign words, and a small number of feminine nouns or singulatives (nomina unitatis) derived from the noun form ʾufʿūlun. Here he indicates that words with the most diverse meanings are formed on this latter pattern and should be strictly kept separated from the invariable ʾufʿūlatun form.

Echoing the preface to the present volume, and thanks to his translation of an unpublished manuscript, that of Hanna Diyab, the aleppine interpreter of doctor Paul Lucas (1664-1737), Elie Kallas invites us, in 'Gerboise : l'entrée du terme arabe ǧerbūʿ à la cour de Louis XIV', to reconsider the origin of the term 'gerboise' (jerboa) in French. Indeed in the description of his travel from Aleppo to Versailles, Diyab provides more details about the introduction of the jerboa, both the animal and the term, at Louis XIV court. Kallas shows here that the introducer might not be Paul Lucas and furthermore that the form of the term in French might be linked to the identity of the jerboa presented to Louis XIV.

In the same vein, but this time for English, Fabrizio A. Pennacchietti's 'On the Semitic Origin of the English Word fustian' addresses the etymology of the English term 'fustian' and its cognates, and explores three possible origins including Arabic, French and Semitic. With evidence, he traces the history of the import of that word and its cognates in English through other languages and shows that only the Semitic origin would be valid. Marie Baize-Varin in '« Traitement » de l'« organisation » en arabe moderne de presse, ou le point de vue d'une linguiste sur l'apparente synonymie ʿilāǧ/muʿālaǧa et tanẓīm/munaẓẓama' revisits an earlier paper of hers and addresses once more the issue of the apparent synonymy existing between pairs like ʿilāǧ/muʿālaǧa and tanẓīm/munaẓẓama. She shows that within these pairs, new occurrences challenge the non-synonymy conclusion based on morphology and indicates that the distinction should be made in terms of cultural considerations, particularly in relation to pairs like specificity / generality and legitimacy / illegitimacy.

In 'Ḥayṯu : une inextricable polysémie ?', after a first pessimistic report on ḥayṯu and biḥayṯu and their inextricable polysemy, Alain Girod aims to explore, thanks to a new corpus consisting of a special issue of the Financial Times devoted to Egypt, written in English but translated into Arabic by al-Maṣrī al-yawm, a surprising track that allows us to account for the multiplicity of uses of this word in the Arabic contemporary press.

Finally, the fourth and last section of this volume, Arabic and Semitic Dialectology, leads to A Ramble into Dialectology. George Grigore presents with 'Fuṣḥā Arabic Vocabulary Borrowed by Mardini Arabic via Turkish' the Mardini Arabic. Spoken in Mardin, a little town in South-eastern Turkey, it has been influenced -at all levels -by the Turkish language, the official language of the area, replacing, for the Arab inhabitants of Mardin, the Fuṣḥā Arabic in all its social functions. This dialect massively borrowed Arabic words from Turkish, which in their turn were borrowed by Turkish from Classical Arabic. These Classical Arabic words entered Mardini Arabic via Turkish language, which gave this Peripheral Arabic dialect an odd image resulting from the mixing of a vocabulary of dialectal and Classical Arabic.

In 'Aspect Marking in Juba Arabic and Ki-Nubi', Kees Versteegh studies the development of aspect markers in an Arabic pidgin, Juba Arabic, spoken in South Sudan, and an Arabic creole, Ki-Nubi, spoken in Kenya and Uganda, both deriving from the 19 th century contacts between the Anglo-Egyptian army in the Sudan and the indigenous recruits. He claims that the earliest varieties of Arabic used in communication in this area did not have any grammaticalized aspectual markers. At a later stage, a general modal marker, BI, was borrowed from Sudanese Arabic. When a second marker, GI, was introduced for the marking of non-punctuality, it took over the marking of habituality from BI.

In 'Jewish Writing in Arabic in Arabic Characters in the Nineteenth and Twentieth Centuries', addressing the issue of Arabic as a medium for other languages as well as the writing of Arabic in other characters, MariaLuisa Langella describes a linguistic, literary and graphic practice consisting of the use of Arabic language and characters by Jews during the 19 th and 20 th centuries. This practice is one aspect of the long-standing relationship between the Jews and the Arabic language, and constitutes a distinctive phenomenon which has so far been little researched. Langella shows that its analysis has revealed the limited scope of the phenomenon from a chronological and geographical point of view, as well as from the point of view of the number of writers concerned, but that, however, these limitations seem to be offset by a certain dynamism, which can be observed through the variety of the types of writings listed.

Introduction ( à

 à Approaches to the History and Dialectology of Arabic in Honor of Pierre Larcher, Manuel Sartori et al. (éds.), Leiden, E. J. Brill, coll.

  In 'The Noun Pattern ʾufʿūlatun in Arabic Philological Tradition', Reinhard Weipert focuses then on another kind of dictionary work referred to as muṣannaf containing sometimes collections of words based on particular noun patterns and not only according to the root of the word. Based on sources like the K. al-Ġarīb al-muṣannaf by ʾAbū ʿUbayd al-Qāsim b. Sallām (d. 224/838), the K. Ǧamharat al-luġa by Ibn Durayd (d. 321/933) and the Dīwān al-ʾadab by al-Fārābī (d. 350/961),

  Turning to synonymy issues, Lidia Bettini focuses in 'La lexicographie arabe entre ʾadab et falsafa : Les questions lexicales du Kitāb al-hawāmil wa-l-šawāmil' on the lexical questions which ʾAbū Ḥayyān al-Tawḥīdī (d. 414/1023) asks ʾAbū ʿAlī Miskawayh (d. 421/1030) in their Kitāb alhawāmil wa-al-šawāmil. What is at stake here is to determine, if possible, the nuance (farq) existing among words perceived as semantically close. After examining the theoretical views of Miskawayh on the question of synonymy, Bettini focuses on five couples of considered synonyms compared to the analogous data offered by authors such as ʿAskarī (d. after 400/1010), Rāġib al-ʾIṣfahānī (d. late 5 th /11 th ) andKaffawī (d. 1094Kaffawī (d. /1683)).

Arkadiusz Płonka's 'Between Linguistics, Poetry, and Ideology: The Literary Periodical L-ʿArzyāda in the Lebanese Language (June 2009-October 2014). General Presentation, Ideological Impacts, Index of Authors, and "Lebanese" Lexis' offers us a presentation of the literary monthly periodical l-ʾArzyāda which is the only journal published in Lebanese since 2009 June 1 st . The journal is a rare and valuable corpus of poetry and prose written by nearly 90 authors. It also contains metalinguistic essays, translations of mainly European poetry and a directory of proverbs and words considered in the journal as typically "Lebanese". Płonka considers the Saʿīd ʿAql's use of the Latin alphabet as a vehicule for writing Arabic, and in particular provides an index of the authors, most of them little-known to specialists of Arabic/Lebanese literature. He also provides a Lebanese language glossary. Finally, Michael G. Carter's 'The Seven Deadly Sins of Arabic Studies' concludes the present volume by discussing a number of features of the study of Arabic linguistics over the last two and a half centuries or so, which, in Carter's opinion, have had negative effects both on our understanding of the Arabs' own grammatical theory and on the teaching and learning of Arabic. After eliminating some well-known topics, namely Flügel's renumbering of the verses of the Qurʾān, the invention of Middle Arabic, and the introduction of the notion of diglossia into pedagogy, the paper looks at seven broad themes which might be considered deadly sins in this context; these, if not corresponding to all seven ethical categories directly, may at least represent the sin of pride. They are (1) transliteration issues, (2) the case and mood names, (3) word classes and parts of speech, (4) verb morphology, (5) definiteness, tanwīn and inflection, with digressions on patterns of definiteness marking, and relative sentences, (6) "government", (7) predication and sentence structure, with digressions on cohesion, and adjectival agreement.

In conclusion, we wish to salute the memory of our colleague Andrzej Zaborski, Professor of Afroasitic linguistics, Jagiellonian University (Krakow), who passed away on September 30, 2014. We take this occasion to express our sadness at losing him, and our respect for his achievements. He was a great creative scholar. He had intended to contribute to the present volume a chapter entitled 'The Verb in Akkadian and in Classical Arabic -Innovation and Archaism', but the was never completed. His provisional abstract read as follows:

Although Akkadian has the oldest records, nevertheless it represents many innovations in phonology, morphology and syntax. Classical Arabic and some non-classical dialects have preserved archaisms which had been lost in Akkadian. Several Akkadian verbal categories are rather innovations both in a Semitic and Hamitosemitic/Afroasiatic perspective and only some of them are retentions. Parallelis between Akkadian iparrVs-u and Ethiopian yeqattel as well as yeqotel of the Modern Semitic of Southern Arabia are only partial. The more diversified system of Classical Arabic is older, e.g. the Arabic (and West Semitic in general) Perfect is not an innovation. Classical Arabic is not 'Jungsemitisch' but 'Old Semitic'. The new relative chronology imposes changes in the genetic classification of the Semitic languages.

Philippe, Manuela, and Manuel


Cristian Dumitrescu
email: cristiand43@gmail.com

Marek Wolf

Marek Wolf The Riemann

The Riemann Hypothesis is not true

come

Introduction

In his only paper devoted to the number theory published in 1859 [START_REF] Riemann | Üeber die Anzahl der Primzahlen unter einer gegebenen Grösse[END_REF] (it was also included as an appendix in [START_REF] Edwards | Riemann's zeta function[END_REF]) Bernhard Riemann continued analytically the series (1) ∞ n=1 1 n s , s = σ + it, σ > 1 to the complex plane with exception of s = 1, where the above series is a divergent harmonic series. He has done it using the integral

(2) ζ(s) = Γ(1 -s) 2πi C (-z) s e z -1 dz z ,
where the contour C is From (3) it is seen that Γ(z) is defined for all complex numbers z, except z = -n for integer n > 0, where are the simple poles of Γ(z). The most popular definition of the gamma function given by the integral Γ(z) = ∞ 0 e -t t z-1 dt is valid only for [z] > 0. Recently perhaps over 100 representations of ζ(s) are known, for review of the integral and series representations see [START_REF] Milgram | Integral and Series Representations of Riemann's Zeta Function and Dirichlet's Eta Function and a Medley of Related Results[END_REF]. The function ζ(s) has trivial zeros at s = -2n, n = 1, 2, 3, . . . and nontrivial zeros in the critical strip 0 < (s) < 1. In [START_REF] Riemann | Üeber die Anzahl der Primzahlen unter einer gegebenen Grösse[END_REF] Riemann made the assumption, now called the Riemann Hypothesis (RH for short in following), that all nontrivial zeros ρ lie on the critical line [s] = 1 2 : ρ = 1 2 + iγ. Contemporary the above requirement is augmented by the demand that all nontrivial zeros are simple.

There are some results on the zero-free regions of ζ(s). K. Ford proved [START_REF] Ford | Zero-free regions for the Riemann zeta function[END_REF], [START_REF] Ford | Vinogradov's Integral And Bounds For The Riemann Zeta Function[END_REF] that ζ(σ + it) = 0 whenever [START_REF] Blanc | A new reason for doubting the Riemann hypothesis[END_REF] σ ≥ 1 -1 57.54(log |t|) 2 3 (log log |t|)

1 3
.

The best present bound seems to belong to M. Mossinghoff and T. Trudgian [START_REF] Mossinghoff | Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function[END_REF]: there are no zeros of ζ(σ + it) for |t| > 2 and

(5)

σ > 1 -1 - 1 5.573412 log |t| .
This is an improvement over (4) as the region free of non-trivial zeros off critical line is larger. In other words the width of the region with possible zeros violating the RH increases with t.

Riemann has shown that ζ(s) fulfills the functional identity:

(6) π -s 2 Γ s 2 ζ(s) = π -1-s 2 Γ 1 -s 2 ζ(1 -s), for s ∈ C \ {0, 1}.
The above form of the functional equation is explicitly symmetrical with respect to the line (s) = 1/2: the change s → 1 -s on both sides of [START_REF] Boros | Sums of arctangents and some formulas of Ramanujan[END_REF] shows that the values of the combination of functions π -s 2 Γ s 2 ζ(s) are the same at points s and 1 -s. Thus it is convenient to introduce the function [START_REF] Borwein | The Riemann Hypothesis: A Resource For The Afficionado And Virtuoso Alike[END_REF] ξ

(s) = 1 2 s(s -1)Γ s 2 ζ(s).
Then the functional identity takes the simple form:

(8) ξ(1 -s) = ξ(s)
The fact that ζ(s) = 0 for (s) > 1 and the form of the functional identity entails that nontrivial zeros ρ n = β n + iγ n are located in the critical strip:

0 ≤ [ρ n ] = β n ≤ 1.
From the complex conjugation of ζ(s) = 0 it follows that if ρ n = β n + iγ n is a zero, then ρ n = β n -iγ n also is a zero. From the symmetry of the functional equation ( 6) with respect to the line [s] =1 2 it follows, that if ρ n = β n + iγ n is a zero, then 1 -ρ n = 1 -β n -iγ n and 1 -ρ n = 1 -β n + iγ n are also zeros: they are located symmetrically around the straight line [s] = 1 2 and the axis t = 0, see Fig. 1. Riemann guesses and von Mangoldt proved that the number of zeta zeros N (T ) with positive imaginary parts < T (and real part inside critical strip) is (see eq.(2.3.6) in [START_REF] Borwein | The Riemann Hypothesis: A Resource For The Afficionado And Virtuoso Alike[END_REF]) ( 9)

N (T ) = T 2π log T 2π - T 2π + 7 8 + O(log(T )).
The classical (from XX century) references on the RH are [START_REF] Titchmarsh | The Theory of the Riemann Zeta-function[END_REF], [START_REF] Edwards | Riemann's zeta function[END_REF], [START_REF] Ivić | The Riemann zeta-function: the theory of the Riemann zeta-function with applications[END_REF], [START_REF] Karatsuba | The Riemann zeta-function[END_REF]. In the XXI century there appeared two monographs about the zeta function: [START_REF] Borwein | The Riemann Hypothesis: A Resource For The Afficionado And Virtuoso Alike[END_REF] and [START_REF] Broughan | Equivalents of the Riemann Hypothesis[END_REF].

There was a lot of attempts to prove RH and the common opinion was that it is true. However let us notice that there were famous mathematicians: J. E. Littlewood [1], P. Turan and A.M. Turing [5, p.1209] believing that the RH is not true. In Karatsuba's talk [START_REF] Karatsuba | [END_REF] at 1:01:10 1 he mentions that Atle Selberg had serious doubts whether RH is true or not. See also the paper "On some reasons for doubting the Riemann hypothesis" [START_REF] Ivic | On some reasons for doubting the Riemann hypothesis[END_REF] (reprinted in [7, p.137]) written by A. Ivić. New arguments against RH can be found in [START_REF] Blanc | A new reason for doubting the Riemann hypothesis[END_REF]. When J. Derbyshire asked A. Odlyzko about his opinion on the validity of RH he replied "Either it's true, or else it isn't" [11, p. 357-358]. There were some attempts to prove RH using the physical methods, see [START_REF] Schumayer | Physics of the Riemann hypothesis[END_REF][START_REF] Wolf | Will a physicist prove the Riemann hypothesis[END_REF]. Some analogs of the RH were proved and some other were disproved. André Weil proved the Riemann hypothesis to be true for field functions [START_REF] Weil | Sur les courbes algebriques et les varietes qui s'en deduisent[END_REF], while the Davenport-Heilbronn zeta-function [START_REF] Davenport | On the Zeros of Certain Dirichlet Series II[END_REF], which shares many properties with usual ζ(s), has zeros outside critical line and even to the right of (s) = 1, see [START_REF] Balanzario | Zeros of the Davenport-Heilbronn counterexample[END_REF].

In this paper we are going to present incompatibility of RH with the theorem on the universality of the logarithmic derivative of the zeta function. In 1975 S.M. Voronin [START_REF] Voronin | Theorem on the Universality of the Riemann Zeta Function[END_REF] proved first universality theorem. He wanted to prove the RH but instead he proved remarkable theorem on the universality of the Riemann ζ(s) function: be a continuous function on K which is holomorphic on the interior of K and is not zero in U ⊂ K. Then for any > 0 there exists a T ( , f ) such that [START_REF] Davenport | On the Zeros of Certain Dirichlet Series II[END_REF]

Voronin's theorem: Let K be a compact subset of the strip D(1/2, 1) = {s ∈ C : 1 2 < (s) < 1} ⊂ C such that the complement of K is connected. Let f : K → C
max |s|≤r f (s) -ζ s + 3 4 + i T ( , f ) < .
Put simply in words it means that the zeta function approximates locally any smooth function in a uniform way.

We will use the universality theorem for the logarithmic derivative of ζ(s) [22, Theorem 2]:

Laurinčikas's theorem: Let K be a compact subset of the strip

D(1/2, 1) = {s ∈ C : 1 2 < (s) < 1} ⊂ C such that the complement of K is connected. Let f : K → C be
an analytic function inside K and continuous up to the boundary of K. Then for any > 0: [START_REF] Derbyshire | Prime Obsession. Bernhard Riemann and the greatest unsolved problem in mathematics[END_REF] lim inf

T →∞ 1 2T µ τ ∈ [-T, T ] : sup s∈K ζ ζ (s + iτ ) -f (s) < > 0 2 Above µ(A)
is the Lebesgue measure of the set A ⊂ R. Let us remark that in Voronin's theorem the existence of just one T such that ζ(s+ iT ) approximates f(s) is guaranteed but in Laurinčikas's theorem the existence of uncountably many such T is secured. Tere are also such "strong" or " enhanced" versions of the Voronin theorem, where the set of T is uncountably, see [START_REF] Lauriničkas | Limit Theorems for the Riemann Zeta-Function[END_REF], [START_REF] Matsumoto | A Survey On The Theory Of Universality For Zeta And L-Functions[END_REF]. In fact, because we assume RH, ζ (s)/ζ(s) is continuous to the right of (s) = 1 . there will be some (short) intervals of T such that ζ (s + iT )/ζ(s + iT ) approximates f (s) -if some T is good then also neighboring T will satisfy |ζ (s + iT )/ζ(s + iT ) -f (s)| < . We stress that in these theorems the RH is not assumed. The idea of our proof is as follows: We assume RH, thus inside any rectangle on the right of critical line there are no zeros of ζ(s). Using the Argument principle and the universality of ζ (s)/ζ(s) we will show that it is possible to choose such rectangles that integral of ζ (s)/ζ(s) along edges of these rectangles will be different from zero meaning that there are zeros of ζ(s) to the right of (s) = 1 2 : contradiction. Thus our main result is:

Thereom: The Riemann Hypthesis is not true.

In the following section we will present proof of above theorem. The Argument Principle. If f (z) is a meromorphic function inside and on some closed contour C, and f has no zeros or poles on C, then

(12) 1 2πi C f f (z) dz = Z -P
where Z and P denote respectively the number of zeros and poles of f (z) inside the contour C, with each zero and pole counted as many times as its multiplicity.

Let us notice that the only pole of ζ(s) is at s = 1. We consider the rectangle ABCD shown in Figure 2. We will denote this rectangle also as D(α, β, T ) where T will be determined later. If we assume that the RH is true then inside this rectangle and on its border there are no zeros of ζ(s), thus the integral We choose f (s) from the Laurinčikas's theorem to be the complex constant on a compact K on the right half of the critical strip (that K can be chosen a horizontal segment):

(15)

F (s) = U + iV, U, V ∈ R.
This F (s) can be constant in fact on the whole complex plane C We can put U = 0 as it will cancels out later. From the Laurinčikas's theorem we know that there exists such T that ( 16)

ζ ζ (s + iT ) = U + iV + r(s)
where the remainder |r(s)| < 1 (later we will use yet other epsilons) on the horizontal side of our rectangle (α + iT, β + iT ), i.e. we choose T as the height of our rectangle ABCD. Thus for segment AB we have (17

) AB ζ ζ (s) ds = (α -β)(U + iV ) + o(1)
where o(1) absorbs term:

(18) AB r(s) ds < (β -α) 1
and can be made arbitrarily small. Because there is a symmetry with respect to real axis:

(19) ζ ζ (s) = ζ ζ (s)
we have (20

) CD ζ ζ (s) ds = (β -α)(U -iV ) + o(1)
and together (21

) AB ζ ζ (s) ds + CD ζ ζ (s) ds = 2i(α -β)V + o(1).
To calculate integrals along vertical segments we will use the following formula for the logarithmic derivative of the ζ(s) function (see e.g. [9, Chapt.12]): [START_REF] Laurinčikas | Zeros of the derivative of the Riemann zeta-function[END_REF] ζ (s)

ζ(s) = 1 1 -s + 1 2 log(π) - 1 2 Γ ( s 2 + 1) Γ( s 2 + 1) + ρ 1 s -ρ .
The last summation extends over all nontrivial zeros of ζ(s), i.e. assuming the RH over ρ = 1 2 ± iγ.

Integrating the first term on rhs of ( 22) we obtain ( 23)

DA 1 1 -s ds = log(β -1 + iT ) -log(β -1 -iT )
We have [START_REF] Matsumoto | A Survey On The Theory Of Universality For Zeta And L-Functions[END_REF] log(z) = log |z| + i arg(z)

hence we obtain ( 25)

DA 1 1 -s ds = 2i arg(β -1 + iT )
Similarly for left vertical segment CB we obtain ( 26)

CB 1 1 -s ds = 2i arg(α -1 + iT )
Together for the positive circulation of the contour ABCD we have ( 27)

DA 1 1 -s ds - CB 1 1 -s ds = 2i(arg(β -1 + iT ) -arg(α -1 + iT ))
The Laurinčikas's theorem assures that T can jump to arbitrarily large T hence:

(28) lim

T →∞ arg(α -1 + iT ) = π 2 lim T →∞ arg(β -1 + iT ) = π 2
and it follows that for sufficiently large T :

(29) DA 1 1 -s ds + BC 1 1 -s ds = o(1).
For the constant term 1 2 log(π) in ( 22) we obtain:

(30) DA 1 2 log(π) ds = 1 2 log(π) T -T i dt = log(π)T i
Adding integral over left vertical segment we obtain:

ds = -(s + 2) + (2 + s) log 1 + s 2 -log (2 + s) β+iT β-iT + O 1 T .
Because 1 + α/2 > 0 and hence also 1 + β/2 > 0, 2 + α > 0, 2 + β > 0 we avoid branch cut in logarithm for negative arguments. After straightforward calculations we obtain for T → ∞ (in fact T does not increase to infinity continuously but T leaps over large values in according with Laurinčikas's theorem):

(37) 1 2 DA Γ ( s 2 + 1) Γ( s 2 + 1) ds + BC Γ ( s 2 + 1) Γ( s 2 + 1) ds T →∞ ------→ (β -α) π 2 i.
Now we will calculate the integrals of the last sum in [START_REF] Laurinčikas | Zeros of the derivative of the Riemann zeta-function[END_REF] over vertical edges of the rectangle ABCD. In the sum (38)

S(t) = ρ 1 s -ρ ,
where s ∈ (α -iT, α + iT ), i.e. s = α + it. The sum over zeros ρ is convergent when zeros ρ and complex conjugate ρ are paired together

(39) 1 s -1 2 -iγ n + 1 s -1 2 + iγ n = 2 s -1 2 (s -1 2 ) 2 + γ 2 n
and summed according to increasing absolute values of the imaginary parts γ n , see [12, Sect.2.5] (it follows also from the M-test of Weierstrass): the sum ρ 1/|ρ -1/2| 1+ over all roots of ζ(s) converges for any > 0. If RH is not true there is a possibility that for some zero ρ denominator s -ρ will be zero and the sum will be infinite. Only when we assume RH ρ = 1 2 + iγ, thus even if for some k there will be t = γ k the denominator will be α -1 2 = 0 and we can have uniformly in t that for any 2 > 0 there exists such N ( 2 ) that ( 40)

n>N ( 2 ) 1 (s -1 2 ) 2 + γ 2 n < 2
for s on the vertical edges of rectangle: s ∈ (α -iT, α + iT ) or s ∈ (β -iT, β + iT ).

To have negligible integrals of the above sum over zeros above along the edges DA and BC we choose

(41) 2 = 1 T 2 .
Thus the remaining sum is finite and we can interchange integration with summation For the vertical CB we write (42)

BC n≤N ( 2 ) 1 s -1 2 -iγ n + 1 s -1 2 + iγ n ds = n≤N ( 2 ) log α - 1 2 -iT -iγ n -log α - 1 2 + iT -iγ n + log α - 1 2 -iT + iγ n -log α - 1 2 + iT + iγ n
The real parts of arguments under logarithms are positive and we use the main branch. Now we make use of ( 24) and the fact that log(|z|) = log(|z|) and arg(z) = -arg(z). Adding similar expression for edge DA we obtain for the summands:

(43) BC 1 s -1 2 -iγ n + 1 s -1 2 + iγ n ds + DA 1 s -1 2 -iγ n + 1 s -1 2 + iγ n ds = 2i arg β - 1 2 + i(T -γ n ) + arg β - 1 2 + i(T + γ n ) -arg α - 1 2 -i(T -γ n ) -arg α - 1 2 + i(T + γ n )
Next we use the relation (44) arg(x + iy) = arctan y x for x > 0 and obtain:

(45)

n≤N ( 2 ) BC 1 s -1 2 + iγ n + 1 s -1 2 -iγ n ds + DA 1 s -1 2 + iγ n + 1 s -1 2 -iγ n ds = 2i n≤N ( 2 ) arctan T -γ n β -1 2 + arctan T + γ n β -1 2 10 -arctan T -γ n α -1 2 -arctan T + γ n α -1 2
We collect separately terms with T + γ n and T -γ n and our sum is following (N = N ( 2 ) for short):

(46)

S N := N k=1 arctan T -γ k β -1 2 -arctan T -γ k α -1 2 + arctan T + γ k β -1 2 -arctan T + γ k α -1 2
Fortunately there is a method of calculating sums of arctan functions by telescoping. Namely we have (see [eq.(2.5)][6]):

Lemma 1: Let f (x) be defined by (47) h(x) = f (x + 1) -f (x) 1 + f (x + 1)f (x) Then (48) n k=1 arctan(h(k)
) = arctan(f (n)) -arctan(f (1)) + π sgn f (k)
where the last sum is taken over all 1 ≤ k ≤ n for which f (k + 1)f (k) < -1.

We have [2, eq. (4.4.34)]

(49) arctan(x) + arctan(y) =

             arctan x + y 1 -xy if xy < 1 arctan x + y 1 -xy + πsgn(x) if xy > 1 where sgn(x) =      1 for x > 0 0 for x = 0 -1 for x < 0 Using it in (46) we obtain terms (50) arctan T -γ k β -1 2 -arctan T -γ k α -1 2 = arctan (α -β)(T -γ k ) (α -1 2 )(β -1 2 ) + (T -γ k ) 2 + πsgn T -γ k β -1 2 , (51) arctan
T + γ k β -1 2 -arctan T + γ k α -1 2 = arctan (α -β)(T + γ k ) (α -1 2 )(β -1 2 ) + (T + γ k ) 2 + πsgn T + γ k β -1 2
Thus we have the sum

(52) S N = N k=1 arctan (α -β)(T -γ k ) (α -1 2 )(β -1 2 ) + (T -γ k ) 2 + N k=1 arctan (α -β)(T + γ k ) (α -1 2 )(β -1 2 ) + (T + γ k ) 2 + Q 1 π
where Q 1 is an integer. We introduce two functions:

(53) h 1 (k) = (α -β)(T -γ k ) (α -1 2 )(β -1 2 ) + (T -γ k ) 2 (54) h 2 (k) = (α -β)(T + γ k ) (α -1 2 )(β -1 2 ) + (T + γ k ) 2
To telescope sums (46) we need to find two functions f (k) and g(k) fulfilling recurrences:

(55)

f (k + 1) = (α -β)(T -γ k ) + (α -1 2 )(β -1 2 )f (k) + (T -γ k ) 2 f (k) (α -1 2 )(β -1 2 ) + (T -γ k ) 2 -(α -β)(T -γ k )f (k) , , (56) g
(k + 1) = (α -β)(T + γ k ) + (α -1 2 )(β -1 2 )g(k) + (T + γ k ) 2 g(k) (α -1 2 )(β -1 2 ) + (T + γ k ) 2 -(α -β)(T + γ k )g(k)
with the initial coditions f (1) = 0, g(1) = 0. We do not need explicit solutions of above equations but behavior of these functions for large k. In the Appendix we will prove the Lemma 2: We have (57) lim

N →∞ f (N ) = ∞ (58) lim N →∞ g(N ) = -∞
Hence:

(59) lim

N →∞ arctan f (N ) = π 2 (60) lim N →∞ arctan g(N ) = - π 2
Larger N ( 2 ) means smaller 2 and smaller term o(1). Finally we obtain:

(61)

S N = Q 1 π + Q 2 π + o(1) = Qπ + o(1)
where Q, Q 1 , Q2 are integers. For the total integral over the rectangle ABCD we get:

(62) 1 2πi ABCD ζ ζ (s)ds = β -α 4 + α -β π V + Q + o(1).
We can choose α and β ( 1 2 < α < β < 1) and V as we wish and we impose condition

(63) β -α 4 + α -β π V = 1 2 .
The term o(1) contains all contributions from epsilon expressions, in particular we choose 2 in (41) such that (64) o(1) < 1 2 .

At this point we reach contradiction: we assumed RH hence the above integral (62) should be zero, but we have impossible inequality with integer Q:

(65) 1 2 = -Q + o(1),
thus RH can not be true.

Final Comments

We assumed RH and using the Laurinčikas's universality theorem we reached the contradiction. In [START_REF] Edwards | Riemann's zeta function[END_REF] the integral on lhs (divided by 2πi) has to be an integer.

In our final expression (62) it seems that this value is not an integer. The same appears in the von Mangoldt formula (9): it also does not take integer values, it is a continuous function of T but there is an error term present which can ensure integer values of the total expression. We also have error term for integral of F (s) = U + iV coming from epsilon in Laurinčikas's theorem. There is an integer Q in (62) and fractional numbers with α -β but the term o(1) can assure that the total value of rhs of ( 62) is integer and equal to the number of zeta zeros inside rectangle ABCD.

There are effective versions of the universality Voronin's Theorem e.g. [START_REF] Garunkštis | Effective uniform approximation by the Riemann zeta-function[END_REF], and obtained numbers are enormously large: they grow like double iterated exponentials. We can expect that T appearing in the Laurinčikas's theorem is also very large. Thus we can say that RH is practically true. Odlyzko has expressed the view that the hypothetical zeros off the critical line are unlikely to be encountered for t below 10 10 10000 , see [11, p.358]. It seems there is no hope to find explicit values of the zeros off the critical line.

In [START_REF] Holt | When Einstein Walked with Goedel : Excursions to the Edge of Thought[END_REF] on pp.60-61 we can read: "Long open conjectures in analysis tend to be false ".

Appendix: prove of Lemma 2

We repeat here in another form equations (55), (??) (66)

f (k + 1) = (T -γ k ) 2 + (α -1 2 )(β -1 2 ) f (k) + (α -β)(T -γ k ) -(α -β)(T -γ k )f (k) + (T -γ k ) 2 + (α -1 2 )(β -1 2 ) , (67) g
(k + 1) = (α -1 2 )(β -1 2 ) + (T + γ k ) 2 g(k) + (α -β)(T + γ k ) -(α -β)(T + γ k )g(k) + (T + γ k ) 2 + (α -1 2 )(β -1 2 )
, with initial conditions f (1) = 0, g(1) = 0. The above equations have the form of the Riccati equation, see e.g [21, Appendix A], [START_REF] Milne-Thomson | TThe Calculus Of Finite Differences[END_REF]:

(68)

x n+1 = a(n)x n + b(n) c(n)x n + d(n) ,
where the denominator cannot be zero. We will first cinsider (66). In our case:

(69)

a(n) = d(n) = (T -γ n ) 2 + (α - 1 2 )(β - 1 2 ), (70) b(n) = -c(n) = (α -β)(T -γ n ).
The customary way to solve the Ricccati equation is to perform transformation of unknown variables in order to get second order linear difference equation. We introduce new variable We also note that (86)

x(n + 1) -x(n) = b(n)(1 + (x(n) 2 ) a(n) -b(n)x(n)
From (85) and above equation we obtain that for sufficiently large n we have (87)

x(n + 1) -x(n) > 0 so for large n the sequence x(n) is increasing. From (68) we see that x(n) can not have real limit, because the equation fulfilled by the limit x * = lim n→∞ x(n)

(88) x * = a(n)x * + b(n) -b(n)x * a(n)
has no real solution. The conclusion is that the sequence x(n) tends to infinity and hence we have also

-

  Appearing in (2) the gamma function Γ(z) has many representations, we present the Weierstrass product: n) = 0.577216 . . . .

Figure 1 :

 1 Figure 1: The location of zeros of the Riemann ζ(s) function.

Figure 2 .

 2 Figure 2. The rectangle D(α, T ), in red critical line is plotted. Sides AB and CD have length β -α while sides CB and DA have length 2T .

  (71)y(n) = H(n)f (n)From Perron's theorem see [26, Chapt.17] we have )H(n) = C From above and (83) we conclude that (85) lim n→∞ |x(n)| |T -γ n | = 0.

  to obtain. Similar calculations lead to the conclusion(90) lim n→∞ g(n) = -∞and the Lemma 2 is proved.

We thank A. Kourbatov for bringing this fact to our attention

where (72)

Then the relation (66) becomes ( 73)

where (74)

Now we perform the second reansformation:

(75)

This change of variables reduces our starting equation (66) to a linear second order equation:

where we have (77)

We note that (78)

so the characteristic equation for the linear linear second order difference equation is:

(79)