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analytic functions arbitrary fixed arithmetic progression assertion basic characters modulo condition Consequently constant implied constant sign Corput critical strip critical zeros Cudakov denote density of zeros distribution of prime E. C. Titchmarsh easy English with Russian equivalent esti extended Riemann hypothesis Farey series formula 9 function f(x function L(s,X function p(x,D,l gives Goldbach-Vinogradov theorem Hardy and Littlewood Hoheisel I. M. Vinogradov Ingham interval a,b Izvestiya Akad K. A. Rodosskii Landau least prime number left side let us remark Linnik gave mean-value theorem method of density Nauk SSSR number of zeros obtain odd primes path of integration positive constant primitive character proof properties prove rectangle relation 15 Renyi right side roots of unity Russian summary Sbornik N. S. sufficiently large summation Suppose symbol theo trigonometrical sums unit circle validity values Vinogradov's theorem well-known whole numbers zeros lying zeros of L-functions