A proof of a conjecture of Bateman and Diamond on Beurling generalized primes

作者：Zhang; Wen Bin

来源：Monatshefte fur Mathematik, 2015, 176(4): 637-656.

DOI：10.1007/s00605-014-0681-8

• Beurling generalized primes
• Prime number theorem
• Wiener-Ikehara tauberian theorem
• Fourier transform
• Schwartz function
• Poisson summation formula

Beurling's generalized prime system is a sequence of real numbers satisfying . The multiplicative semigroup generated by is called a system of Beurling's generalized integers. If and denote the counting functions of generalized primes and of generalized integers , respectively, Beurling's problem is to find conditions on which imply "the prime number theorem", i.e., as . Assuming that , Beurling's condition is with ; but does not suffice. Bateman and Diamond conjectured the condition . A proof of this conjecture following Kahane's method is given in this paper. The proof is based on Poission's summation formula and the Wiener-Ikehara tauberian theorem and applies classical ideas.

• 出版日期2015-4
• 单位华南理工大学