Research Projects

For a periodic arithmetic function with period $q,$ and a complex number $s$ with real part greater than zero Dirichlet introduced the series in 1837 popularly known as the Dirichlet series or $L$-function associated to the function $f$. This series admits the analytic continuation to the whole complex plane except at $s=1$ where it has a simple pole. The arithmetic nature of the special values of the $L$-function has been the focus of study for a long time. Many mathematicians have worked in this direction and they have got some successful results. One of the important problem was raised by Chowla well known as the Chowla's conjecture in 1969 regarding the the existence of some special periodic function. This was solved by Baker, Birch, Wirsing in 1973 using an application of a theorem of Baker concerning linear forms in logarithms of algebraic numbers. Also the conjecture made by Erd\"{o}s in 1973 regarding the non-vanishing of Dirichlet series has almost been successfully solved by many mathematicians but still it is open for some special case. For $s=1$ and $f$ being a non-trivial Dirichlet character, Dirichlet proved that the series is non-zero and hence a transcendental number by the famous theorem of Baker regarding the transcendental nature of the algebraic linear combination of logarithm of algebraic numbers. There are some remarkable results in the linear independence of these series for the non-trivial distinct Dirichlet characters over some specific fields by Ram Murty, Baker, Birch, Wirsing and Saradha. We will study the special values of the derivatives of these Dirichlet $L$ functions. We also investigate special values of p-adic L-functions/ p-adic analogue of classical constant and q-analogues of various classical constant.