In the 1980’s author proved lower bounds for mean value of modulus error term prime number theorem and other important theoretic functions whose oscillation is in connection with zeros Riemann zeta function. present work a general shown simple way which gives bound mentioned as function hypothetical pole Mellin transform The conditions are amply satisfied such results recover earlier ones (even...

By Grothedieck's Anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number fields encode all arithmetic information of these curves. The goal of this paper is to develope and arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing th...