Let a be a natural number different from 0. In 1963, Linnik proved the following unconditional result about the Titchmarsh divisor problem [Formula: see text] where c is a constant dependent on a. Titchmarsh proved the above result assuming GRH for Dirichlet L-functions in 1931. We establish the following asymptotic relation: [Formula: see text] where Ck is a constant dependent on k and a, and the implied constant is dependent on k. We also apply it a question related to Artin's conjecture for primitive roots.

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Bipolar fuzzy graph is more precise than a fuzzy graph when dealing with imprecision as it is focusing on the positive and negative information of each vertex and edge. Nowadays, researchers have utilized bipolar fuzzy graphs in decision-making problems. Bipolar fuzzy competition graphs aid to compute the competition between the vertices in bipolar fuzzy graphs. To depict the best competitions among the competitions of bipolar fuzzy graphs, the best bipolar fuzzy competition graph can be defined using bipolar fuzzy α-cut and the strength of the competition between the vertices can also be determined. Fuzzy graphs are used well to frame modelling in real-time problems. In particular, when the real-time scenario is modelled using the bipolar fuzzy graph, it gives more precision and flexibility. At present, researchers have focused on decision-making techniques with bipolar fuzzy graphs. The DEMATEL method is one of the powerful decision-making tools. It effectively analyses the complicated digraphs and matrices. The fuzzy DEMATEL technique can convert the interrelations between factors into an intelligible structural model of the system and divide them into cause and effect groups. Therefore, this study attempts to design the DEMATEL method under the bipolar fuzzy environment. To illustrate this proposed technique, the problem of identifying the best mobile network is taken. With this method, the benefits and drawbacks of networks are measured and a complicated bipolar fuzzy directed graph can be transformed into a viewed structure.

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