By K. Ramachandra

"Theory of Numbers: A Textbook" is aimed toward scholars of arithmetic who're graduates or maybe undergraduates. little or no necessities are wanted. The reader is predicted to grasp the speculation of features of a true variable and in a few chapters advanced integration and a few basic rules of advanced functionality concept are assumed. the complete booklet is self contained other than theorems 7 and nine of bankruptcy eleven that are assumed. the main formidable bankruptcy is bankruptcy eleven the place the main beautiful end result on distinction among consecutive primes is proved. References to the most recent advancements like Heath-Brown's paintings and the paintings of R.C. Baker, G. Harman and J. Pintz alongwith readable debts of Brun's sieve and in addition of Apery's Theorem on irrationality of zeta (3) are given. eventually the reader is conversant in Montgomery-Vaughan Theorem within the final bankruptcy. it's was hoping that the reader will benefit from the leisurely form of presentation of many very important effects.

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D. CHOWLA, P. ERDOS, K. MAHLER, M. PARTHASARATHY, L. J. LANDER, T. R. PARKIN and J. L. SELFRIDGE are mentioned as references in this paper. Dr. R. html According to this the largest prime number known is 2 3 0 4 0 2 4 5 7 - 1. It contains more than 91 lakh digits. Other facts are mentioned in the URL. Chapter 2 SIMPLE Q RESULTS BASED ON SIMPLE PROPERTIES CM §LThe arithmetical function A(n). n, n. (i+*) Our object is to prove the following Theorem 1. We have lim sup [ x'1'2 |^ n—x 29 A(n)| I > 0.

H Proof. We can obviously assume that h ^ x. Consider Q(x + h)-Q{x)= £ x

The same conclusions follow from our next proof also. Second Proof. - = g 1-T ^ = f/9n(s) x n=l n=l say, where 9nK > n° Jn u* Jn \n° Wj Jn \Jn v'+ij We have which shows that 5^1° gn (s) converges uniformly in any fixed half-plane a ^ S > 0. ). (4)... n=l U ^ ^ Prom this, it follows immediately that lim5_>i+o ((s - l)C(s)) = 1- Also, we get a \s-\\ so that for 0 < s < 1, , C W , > _ 1 + _L- = _£_ > 0 . Thus we have proved the Lemma. Proof of Theorem 1. Suppose that we have A (")i ^ e *1/2 i£ n—i for some e > 0 and for all x ^.