Topics in prime number theory

The thesis is divided into five sections: (a) Trigonometric sums involving prime numbers and applications, (b) Mean-values and Sign-changes of S(t)-- related to Riemann's Zeta function, (c) Mean-values of strongly additive arithmetical functions, (d) Combinatorial identities and sieves, (e) A Goldbach-type problem. Parts (b) and (c) are related by means of the techniques used but otherwise the sections are disjoint. (a) We consider the question of finding upper bounds for sums like ∑_PSN▒〖e(ap2)〗 and using a method of Vaughan, we get estimates which are much better than those obtained by Vinogradov. We then consider two applications of these, namely, the distribution of the sequence (αp2) modulo one. Of course we could have used the improved results to get improvements in estimates in various other problems involving p[superscript]2 but we do not do so. We also obtain an estimate for the sum ∑_PSN▒〖(ap3)〗 and get improved estimates by the same method. (b) Let N(T) denote the number of zeros of ς(s) - Riemann's Zeta function. It is well known that N(T) = L(T) + S(T), where L(T) = 1/2π Tlog(T/2π)-T⁄(2π+7⁄(8+0 ((1)⁄(T))))but the finer behaviour of S(T) is not known. It is known that S(t) ≪ log t ; ∫_o^t▒〖Slu)du〗 ≪ log t so that S(T) has many changes of sign. In 1942, A. Selberg showed that the number of sign changes of Set) for t ∈ (O,T) exceeds T (log T)1/3 exp(-A loglog T), (1) but stated to Professor Halberstam in 1979 that one can improve the constant 1/3 in (1) to 1 – ∈. It can be shown easily that the upper bound for the number of changes of sign is log T. We give a proof of Selberg's statement in (b), but in the process we do much more. Selberg showed that if k is a positive integer then ∫_T^(T+H)▒〖ls(t)l〖2k〗_dt 〗 = C CkH(loglog T) k ,{1+0( (loglogT)(-1)/2) } (2) where TT 1⁄2< H ≤ T[superscript]2 and C[subscript]k is some explicit constant in k. We have found a simple technique which gives (2) with the constant k replaced by any non-negative real number. Using this type of result, I prove Selberg's statement, with (log T)-∈ replaced by Exp (-A√loglogT (logloglogT) -□(1/2)). (c) I use the" method for finding mean-values above to answer similar questions for a class of strongly additive arithemetical functions. We say that f is strongly additive if (1) f(mn) = f(m) = f(n), if m and n are coprime, (2) f(p[superscript]a) = f(p) for all primes p and positive integer a. (d) This section contains joint work with Professor Halberstam and is still in its infancy. We have found a general identity and a type of convolution which serves to be the starting point of most investigations in Prime Number Theory involving the local and the global sieves. The term global refers to sieve methods of Brun, Selberg, Rosser and many more. The term local refers to things like Selberg's formula in the elementary proof of the prime number theorem, Vaughan's identity and so on. We have shown that both methods stem from the same source and so leads to a unified approach to such research. (e) I considered the question of solving the representation of an integer N in the form N = P_(1^2 )+ P _(2^2 ) +K P[subscript] 3, where the Pi’s are prime numbers. This problem was motivated by Goldbach's Problem and is exceedingly difficult. So I looked into getting partial answers. Let E(x) denote the numbers less than x not representable in the required form. Then there is a computable constant δ > 0 such that E(x) ≪ X i- δ To do this we use a method of Montgomery and Vaughan but the proof is long and technical, and we do not give it here. We show by sieve methods that the following result holds true: N = P_(1^2 ) + P _(2^2 ) +kP3P4P5. We have been unable to replace the product of three primes by two. Note: k is a constant depending on the residue class of N modulo 12