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We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$\tau(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $\tau(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations., Comment: To appear in JNT Prime. For more general results, see arXiv:2005.10354

Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$ approaches an even order $2k$ root of unity $\zeta$, \[\lim_{q\to \zeta} \big(f(q) - (-1)^k (1-q)(1-q^3)(1-q^5)\cdots (1-2q + 2q^4 - \cdots)\big) = O(1),\] and hinted at the existence of similar statements for his other mock theta functions. Recent work of Folsom-Ono-Rhoades provides a closed formula for the implied constant in this radial limit of $f(q)$. Here, by different methods, we prove similar results for all of Ramanujan's 5th order mock theta functions. Namely, we show that each 5th order mock theta function may be related to a modular bilateral series, and exploit this connection to obtain our results. We further explore other mock theta functions to which this method can be applied., Comment: 15 Pages

Sayılar Teorisi denince akla gelen bilim insanlarından birisi olan Srinivasa Ramanujan sadece yaşadığı yüzyıla değil modern matematiğin tarihi zirve yaptığı 21. yüzyıla da süren etkisiyle çok önemli başarılara imza atmıştır. Altı bölümden oluşan bu çalışmada Ramanujan'ın sonuçlarına temel oluşturan Ramanujan tau fonksiyonu çalışılmıştır. Modüler formların tanıtılması ile başlayan çalışmanın ikinci bölümünde Ramanujan tau fonksiyonu tanımlanıp temel özellikleri incelenmiştir. Üçüncü bölümde parçalanış fonksiyonu verilmiş ve Ramanujan'ın bu fonksiyon üzerine önemli sonuçlar verdiği kongrüanslar incelenmiştir. Dördüncü bölüme Ramanujan konjektürleri incelenmiştir. Beşinci bölümde ise Ramanujan tau fonksiyonun Fourier katsayılarının asallığı üzerine yapılan çalışmalar üzerinde durulmuştur. Çalışmanın altıncı ve son bölümünde ise Ramanujan tau fonksiyonuna dair güncel bir çalışma incelenmiştir. Çalışma derleme niteliğindedir. Srinivasa Ramanujan, one of the scientists that comes to mind when it comes to NumberTheory, has achieved significant success not only in the century he lived in, but also in the 21st century, when modern mathematics reached its historical peak. In this study, which consists of six chapters, the Ramanujan tau function, which forms the basis for Ramanujan's results, is studied. In the second part of the study, which started with the introduction of modular forms, the Ramanujan tau function was defined and its basic properties were examined. In the third chapter, the partition function is given and the congruences on which Ramanujan gives important results on this function are examined. In the fourth chapter, Ramanujan conjectures are examined. In the fifth chapter, the studies on the primality of the Fourier coefficients of the Ramanujan tau function are emphasized. In the sixth and last part of the study, a recent study on the Ramanujan tau function was examined. The study is a compilation.

In this paper, we study monotone set functions defined as the composition of an additive measure with a strictly increasing function. This function is a unary operator in continuous-valued logic, called the tau function, and it is a generator function-based parametric mapping. We provide a necessary and sufficient condition for the equality of two tau functions that are induced by different generator functions. Using the tau function and its properties, we introduce a new monotone measure that we call the tau-additive measure. This measure is computationally simple and it can be viewed as an upper or lower probability depending on the parameter settings of the tau function. We present the parameter-dependent submodularity and supermodularity of the tau-additive measure and show how this measure can be constructed from a set function on a finite set. This procedure is analogous to how the well-known lambda-additive measure can be constructed, but our method is computationally simpler. We demonstrate that the tau-additive measure can be used to approximate the lambda-additive measure. Lastly, exploiting these theoretical results, we present an application in the area of human resource management.