Vinogradov's Integral and Bounds for the Riemann Zeta Function
Vinogradov's Integral and Bounds for the Riemann Zeta Function
复制标题
黎曼 Zeta 函数的维诺格拉多夫积分和界限
DOI：
10.1112/s0024611502013655
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发表时间：
2002
影响因子：
1.8
通讯作者：
Kevin Ford
中科院分区：
文献类型：
--
作者：
Kevin Ford
主要结果是Riemann Zeta函数在临界带上的一个上界：ζ(σ+it)⩽A|t|B(1−σ)3/2log2/3|t|，其中A=76.2，B=4.45，对1/2⩽σ⩽1和|t|⩾3有效。工具包括有界指数和的Korobov-Vinogradov方法的变体，Vinogradov积分的T.D.Wooley界的显式版本，以及没有小素数因子的数的指数和的平均值的显式界，也使用Wooley的方法。一个辅助结果是指数和界S(N，t)⩽9.463N1−1/(133.66λ2)，其中N是正整数，t是实数，λ=LOG(T)/(LOG N)并且S(N，t)=max 0
The main result is an upper bound for the Riemann zeta function in the critical strip: ζ(σ+it)⩽A|t|B(1−σ)3/2log2/3|t| with A = 76.2 and B = 4.45, valid for ½ ⩽ σ ⩽ 1 and |t| ⩾ 3. The previous best constant B was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound S(N,t)⩽9.463N1−1/(133.66λ2) , where N is a positive integer, t is a real number, λ = log (t)/(log N) and S(N,t)=max0