Using Hilbert’s inequality, we give a new asymptotic formula (uniform inq andT) for$$\mathop \Sigma \limits_{\begin{array}{*{20}c} {\chi (mod q)} \hfill \\ {\chi primitive} \hfill \\ \end{array} } \smallint _T^{2T} |L(\tfrac{1}{2} + it,\chi )^4 |dt$$

For 0 < α < 1, letζ(s, α) be the Hurwitz zeta function and let ζ₁ (s, α) = ζ(s, α) -α− s. For a fixeds, we developζ₁(s,α) as a power series in α in the complex circle ¦α¦ < 1. If$$\sum\limits_{\chi \left( {\bmod q} \right)} {L\left( {s,\chi } \right)L\left( {s',\bar \chi } \right)} = \frac{{\phi \left( q \right)}}{{q^{s + s'} }}\sum\limits_{k/q} \mu \left( {\frac{q}{k}} \right)\left( {\sum\limits_{a = 1}^k {\left( {\frac{k}{a}} \right)^{\operatorname{Re} s + \operatorname{Re} s'} + Q\left( {s,s',k} \right)} } \right)$$, we obtain an asymptotic expansion for Q(k) =Q(s,s′,k) using the power series forζ₁(s,α)

Following appropriate use of approximate functional equation for Hurwitz Zeta function, we obtain upper bounds for$$s(\sigma + it) = \sum\limits_{x^{(modq)} } {|L(\sigma + it,x} )|^2 $$} Here fors = σ + it, L(s,x) denotes DirichletL-series for character x(modq). In particular, we obtain S(1/2 +it) ≪q logqt + t^5/8 q^−1/8, which is an improvement in the range q ¦t¦ < q^11/7, on hitherto best known result. This incidentally gives S(1/2+ it)≪ q log3q for ¦t¦q^9/5.

In the famous work of Dirichlet on class number formula, $L(s, \chi)$ at $s=1$ has been expressed as a finite sum, where $L(s, \chi)$ is the Dirichlet 𝐿-series of a real Dirichlet character. We show that this expression with obvious modification is valid for the general primitive Dirichlet character 𝜒.