Abstract. In this article, we report on computations that led to the discovery of a new Lehmer pair of zeros for the Riemann ζ function. Given this new close pair of zeros, we improve the known lower bound for de Bruijn-Newman constant Λ. The Riemann hypothesis is equivalent to the assertion Λ ≤ 0. In this article, we establish that in fact we have Λ > −1.14541 × 10 −11 . This new bound confirms the belief that if the Riemann hypothesis is true, it is barely true.
Abstract. In this article, we study the problem of changes of sign of π(x) − li(x). We provide three improvements. First, we give better esimates of error term for Lehman's theorem. Second, we rigorously prove the positivity of this difference for a region formerly conjectured by Patrick Demichel. Third, we improve the estimates for regions of positivity by using number theoretic results.
An analysis of the local variations of the prime counting function π ( x ) \pi (x) due to the impact of the non-trivial, complex zeros ϱ k \varrho _k of ζ ( s ) \zeta (s) is provided for x > 10 10 13 x>10^{10^{13}} using up to 200 billion ζ ( s ) \zeta (s) complex zeros. A new bound for | l i ( x ) − π ( x ) | > x 1 / 2 ( l o g l o g l o g x + e + 1 ) / e l o g x |\mathrm {li}(x)-\pi (x)|>x^{1/2}(\mathrm {log}\,\mathrm {log}\,\mathrm {log}\,x+e+1)/e\, \mathrm {log}\,x is proposed consistent with the error growth rate in Littlewood’s proof that l i ( x ) − π ( x ) \mathrm {li}(x)-\pi (x) changes sign infinitely often. This bound is also consistent with all presently known cases where π ( x ) > l i ( x ) \pi (x)>\mathrm {li}(x) including many new examples listed. This implies that Littlewood’s constant K = 1 / e \mathrm {K}=1/e , the lower bound for Skewes’ number is 3.17 × 10 114 3.17\times 10^{114} and the positive constant c c in the Riemann Hypothesis equivalent | l i ( x ) − π ( x ) | > c l o g ( x ) x 1 / 2 |\mathrm {li}(x)-\pi (x)|>c\,\mathrm {log}(x)x^{1/2} is less than 3 × 10 − 27 3\times 10^{-27} .
Making the right platform choice has always been a challenge for the HPC users no matter the applications vertical they are in. The number of references is very large and making the wrong choice can have adverse effects. Formerly users only had to choose between, for example, the different processors and interconnect vendors. Lately, due to the new Intel Skylake processors the choice has become increasingly difficult as different levels of performance are available within the same vendor platforms. To facilitate selection and give possible directions for the real benchmarked applications we introduce the Kernel Generator, an open source tool generating assembly kernels to help the programmer or the benchmarker understand the behavior of the different micro-architectures. We used our tool to study the behavior of the current micro-architectures and compare it to the current synthetic benchmarks which sometimes are not correctly characterizing a platform nor expose its strengths. The Kernel Generator facilitates the discovery of the platforms performance fit. To insure the relevance of our kernel, we are looking at Ansys Fluent behavior to explain the performance on the different Intel processors. In this case, we have that 4100 and 6100 Intel processors families can have equivalent performance on codes not well vectorized: Fluent being one of them. This demonstrates that we can use our tool for initial profiling and understanding of the different platforms.
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