Prime number theoremRosser’s theorem11A41In 1845, Bertrand conjectured that for all integers $x\\\ge2$, there exists at least one prime in $(x/2, x]$. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any $n\\\ge1$, there is...
Heath-Brown, D.R.: The Pjateckiĭ–S̆apiro prime number theorem. J. Number Theory 16(2), 242–266 (1983) MathSciNet Google Scholar Helfgott, H.A.: Major arcs for Goldbach’s problem, preprint (2013) Helfgott, H.A.: The ternary Goldbach problem. Rev. Mat. Iberoam. 130(2)...
explicit formulasprime number theoremIn this paper, explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized....
mathematician of the highest quality, a man of altogether exceptional originality and power."[56] One colleague, E. H. Neville, later commented that "not one [theorem] could have been set in the most advanced mathematical examination in the world."[57] On 8 February 1913, Hardy wrote a l...
On the Prime Number Theorems A few years ago, Watson succeeded in proving an asymptotic congruence property formulated by Ramanujan, namely, the following theorem: If m and Ic are fixed positive integers, there are between n = 1 and n = N only o (N) integers n for w... A Wintner -...
Explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized. 文档格式: .pdf ...
Understanding these sums and their distribution is an important topicof study in number theory, with profound connections to problems in arithmetic such as in theproof of Vinogradov’s theorem [16, Chapter 8], Waring type formulas [12], distribution of rationalnumbers in short intervals [9], ...
if j(τ)j(τ) is the elliptic modular function, then one can explicitly evaluate the value k(τ)k(τ) (Theorem 1.3), and the other is that once the value k(τ)k(τ) is given, we can obtain the value k(rτ)k(rτ) for any positive rational number r immediately (Theorem 1.4)...
summation. On the other hand, he knew nothing of doubly periodic functions, the classical theory of quadratic forms, or Cauchy’s theorem, and he had only the mostnebulousidea of whatconstitutesa mathematical proof. Though brilliant, many of his theorems on the theory of prime numbers were ...
For example, let the denominators and the numerators of the rational number 𝑐𝑛cn be 𝐷𝑛,𝑁𝑛Dn,Nn, respectively. We have the following properties ([6], Theorem 1.1): The denominator 𝐷𝑛Dn is a square-free integer. The set of the all odd prime divisors of 𝐷𝑛Dn ...