素数的黎曼磁子
The Riemann Magneton of the Primes
摘要：
我们提出了一个计算涉及的函数的黎曼Zeta函数和建议的两个最近的作品有关的黎曼假设：一个由Balazard，Saias和Yor和其他Volchkov。我们定义了一个包含Zeta函数的复变函数s = r + it的积分m（r），并找到了一个特别有趣的m（r）的表达式，它至少在r的某些范围内是严格的。在这样的范围内，我们发现导数m '（r）在r = 1和r = 0处有两个间断，我们精确地计算了这两个间断。在r = 1处的跳跃由4*Pi给出，而在r = 0处的跳跃由（-4 + gamma + 3*log（2）+ Pi/2）*Pi给出。对于m（r）直到r = 1/2的表达式的有效性等同于黎曼假设（RH）的真实性。假设RH，m（r）的表达式在r = 1/2时给出m = 0，并且在r = 1/2时斜率m '（r）= Pi*（1 + gamma）= 4.95（其中gamma = 0.577215.是欧拉常数）。因此，如果m（r）的表达式可以一直延续到r = 1/2，那么如果我们将m（r）解释为带电细丝的有效电势，在所有零点处都存在单位电荷的情况下（平凡和非平凡零点）并且在Riemann Zeta函数之一的极点处，则存在电场E（r）= m '（r）在r = 1/2由2*Pi乘以第一Li的系数给出，λ da_1 = [1+gamma/2-（1/2）ln（4*Pi）] = 0.0230957。然后，我们构造一个势阱（x = r+1/2 = 1周围的对称函数），如果RH为真，则该势阱是精确的。与RH无关，通过观察m（r）在r = 1-时的收敛泰勒展开的行为，可以使用欧拉乘积公式来计算m（r = 1/2+）值以及第一个Li系数。我们用这种方法进一步证明了黎曼猜想可能是真的。
英文摘要：
We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m (r) involving the Zeta function in the complex variable s = r + it and find a particurarly interesting expression for m(r) which is rigorous at least in some range of r. In such a range we find that there are two discontinuities of the derivative m'(r) at r = 1 and r = 0, which we calculate exactly. The jump at r = 1 is given by 4*Pi, while that at r = 0 is given by (-4 + gamma + 3*log(2) + Pi/2)*Pi. The validity of the expression for m(r) up to r = 1/2 is equivalent to the truth of the Riemann Hypothesis (RH). Assuming RH the expression for m(r) gives m = 0 at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) = 4.95 at r = 1/2 (where gamma = 0.577215... is the Euler constant). As a consequence, if the expression for m(r) can be continued up to r = 1/2, then if we interpret m(r) as the effective potential of a charged filament, in the presence of the unit charges located at all the zeros (trivial and non trivial zeros) and at the pole in one of the Riemann Zeta function then there is a jump of the electric field E(r) = m'(r) at r = 1/2 given by 2*Pi times the first Li's coefficient lambda_1 = [1+gamma/2-(1/2)ln(4*Pi)] = 0.0230957. We then construct a potential well (a symmetric function around x = r+1/2 = 1) which is exact if the RH is true. Independently of the RH, by looking at the behaviour of the convergent Taylor expansion of m(r) at r = 1-, the value m(r = 1/2+) as well as the first Li's coefficient may be evaluated using the Euler product formula. We give in this way further evidence for the possible truth of the Riemann Hypothesis.
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