On prime primitive roots
On q-orders in primitive modular groups
We prove an upper bound for the number of primes p ≤ x in an arithmetic progression 1 (mod Q) that are exceptional in the sense that has no generator in the interval [1,B]. As a consequence we prove that if with a sufficiently large absolute constant c, then there exists a prime q dividing Q such that for some positive integer b ≤ B. Moreover we estimate the number of such q’s under suitable conditions.
On Ramanujan's formula for values of Riemann zeta-function at positive odd integers
On Riemann's Zeta Function.
On sign-changes in the remainder-term of the prime-number formula, III
On some combinatorial relations for Tornheim's double series
On some conjectures and results for the Riemann zeta-function and Hecke series
On Some Integrals Involving the Mean Square Formula for the Riemann Zeta-function
On some mean value results for the zeta-function in short intervals
Let denote the error term in the Dirichlet divisor problem, and let E(T) denote the error term in the asymptotic formula for the mean square of |ζ(1/2+it)|. If E*(t) := E(t) - 2πΔ*(t/(2π)) with Δ*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x) and , then we obtain a number of results involving the moments of |ζ(1/2+it)| in short intervals, by connecting them to the moments of E*(T) and R(T) in short intervals. Upper bounds and asymptotic formulae for integrals of the form ∫T2T(∫t-Ht+H |ζ(1/2+iu|2 duk dtare...
On some mean value results involving |zeta(1/2+ it)|
On some mean value results involving .
On some problems involving Hardy’s function
Some problems involving the classical Hardy function , are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.
On Some Relations which are Equivalent to Functional Equations Involving the Riemann Zeta Function.
On some theorems of Littlewood and Selberg, IV
On the application of Turán's method to the theory of Dirichlet L-functions
On the asymptotic formulas for powerful numbers.
On the asymptotics of the number of reduced integral quadratic forms with a condition of divisibility of the first coefficients.
On the average behavior of the Fourier coefficients of th symmetric power -function over certain sequences of positive integers
We investigate the average behavior of the th normalized Fourier coefficients of the th ( be any fixed integer) symmetric power -function (i.e., ), attached to a primitive holomorphic cusp form of weight for the full modular group over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum where is sufficiently large, and When , the error term which we obtain improves the earlier known result.
On the average of inversion of Dirichlet’s -functions
On the Brun-Titchmarsh theorem
The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size when logx/logq ≥ 8 and is bounded....