International Mathematics Research Notices

doi: 10.1093/imrn/rnq278pmid: N/A

doi: 10.1093/imrn/rnq276pmid: N/A

doi: 10.1093/imrn/rnq277pmid: N/A

Koukoulopoulos, Dimitris

doi: 10.1093/imrn/rnq045pmid: N/A

We bound from below the number of shifted primes p + s x that have a divisor in a given interval (y,z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants.

Manchon, Dominique; Paycha, Sylvie

doi: 10.1093/imrn/rnq027pmid: N/A

We define renormalized nested sums of symbols on which satisfy stuffle relations. For appropriate symbols, these give rise to renormalized Euler-Zagier-Hoffman multiple zeta (and Hurwitz zeta) functions which satisfy stuffle relations at all arguments. We show the rationality of renormalized multiple zeta values at non-positive integer arguments. These results generalize to radial symbols on n giving rise to a higher-dimensional analog of multiple zeta functions.