We obtain a series of new conditional lower bounds for the modulus and the argument of the Riemann zeta function on very short segments of the critical line, based on the Riemann hypothesis. In particular, we prove that for any large fixed constant A > 1 there exist(non-effective) constants T₀(A) > 0 and c₀(A) > 0 such that the maximum of |ζ (0.5+it)| on the interval (T-h,T+h) is greater than A for any T > T₀ and h = (1/π)lnlnln{T}+c₀.

Let $p$ be an odd prime and $c$ a fixed integer with $(c,p)=1$. For each integer $a$ with $1\le a\le p-1$, it is clear that there exists one and only one $b$ with $0\le b\le p-1$ such that $ab\equiv c$ (mod $p$). Let $N(c,p)$ denote the number of all solutions of the congruence equation $ab\equiv c$ (mod $p$) for $1\le a$, $b\le p-1$ in which $a$ and $\overline{b}$ are of opposite parity, where $\overline{b}$ is defined by the congruence equation $b\overline{b}\equiv 1(modp)$. The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet $L$-functions to study the hybrid mean value problem involving...

The distribution of the vector (|ζ(s, α)|; ζ(s, α)), where ζ(s, α) is the Hurwitz zeta-function with transcendental parameter α, is considered and a probabilistic limit theorem is obtained. Also, the dependence between |ζ(s, α)| and ζ(s, α) in terms of m-characteristic transforms is discussed.

AMS Subj. Classification: 11M41, 11M26, 11S40We study the generalized Li coefficients associated with the class S♯♭ of functions containing the Selberg class and (unconditionally) the class of all automorphic L-functions attached to irreducible unitary cuspidal representations of GLN(Q) and the class of L-functions attached to the Rankin-Selberg convolution of two unitary cuspidal automorphic representations π and π′ of GLm(AF ) and GLm′ (AF ). We deduce a full asymptotic expansion of the Archimedean...

For a fixed integer $k\ge 3$, and fixed $\frac{1}{2}\<\sigma \<1$ we consider$${\int }_1^T{\left|\zeta (\sigma +it)\right|}^{2k}dt=\sum _{n=1}^{\infty }{d}_k^2\left(n\right)n^{-2\sigma }T+R(k,\sigma ;T),$$where $R(k,\sigma ;T)=0\left(T\right)(T\to \infty )$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k,\sigma ;T)$ are derived in the range min $({\beta }_k,{\sigma }_k^{*})\<\sigma \<1$. We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

In this paper, we prove multiple analogues of famous Ramanujan’s formulas for certain Dirichlet series which were introduced in his well-known notebooks. Furthermore, we prove some multiple versions of analogous formulas of Ramanujan which were given by Berndt and so on.

In this article we prove a trace formula for double sums over totally hyperbolic Fuchsian groups $\Gamma$. This links the intersection angles and common perpendiculars of pairs of closed geodesics on $\Gamma /H$ with the inner products of squares of absolute values of eigenfunctions of the hyperbolic laplacian $\Delta$. We then extract quantitative results about the intersection angles and common perpendiculars of these geodesics both on average and individually.