On a Mean Value of Gadiyar and Padma

2015/06/24 by John Washburn, Washburn, John
Mathematics · #11A41 #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM)

paper · pdf · doi:10.48550/arxiv.1506.07822

openalex publication_date 2015/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Building on the earlier works of Gadiyar and Padma, the main result of this paper is to prove: lim_{n → ∞} (1)/(N) ∑_n=1^N (ϕ(n) Λ(n ))/(n) (ϕ(n+h) Λ(n +h))/(n+h) = ∑_q=1^∞ \Vert (μ(q))/(ϕ(q)) \Vert² c_q(h) This sieve with Ramanujan-Fourier expansions is the the central relationship to be proven in within the works of H. G. Gadiyar and R. Padma, as related to the following conjectures in number theory: The twinned prime conjecture, The Sophie Germaine Primes conjecture, and Conjectures B and D of Hardy and Littlewood. A reviewer has point out that Theorem 8 from the previous version should be split into two theorems; one for absolute convergence and one for uniform convergence.

Citations

• Ramanujan - Fourier Series and the Density of Sophie Germain Primes
• Linking the Circle and the Sieve: Ramanujan - Fourier Series
• Rota meets Ramanujan: Probabilistic interpretation of Ramanujan - Fourier series
• The Wiener-Khinchin Theorem for Non-wide Sense stationary Random Processes

Related

• An Entry of Ramanujan on Hypergeometric Series in his Notebooks
• A smooth summation of Ramanujan expansions
• On Ramanujan expansions and primes in arithmetic progressions
• Some asymptotic formulae involving Cohen-Ramanujan expansions
• On an identity of Delange and its application to Cohen-Ramanujan expansions