{"version":9,"randomSeed":"2162ef36cd4e4799bbce12c744f96046","graph":{"viewport":{"xmin":-9.999999999999998,"ymin":-10,"xmax":9.999999999999998,"ymax":9.999999999999996}},"expressions":{"list":[{"type":"text","id":"14","text":"https://en.wikipedia.org/wiki/Riemann_zeta_function#Globally_convergent_series"},{"type":"text","id":"3","text":"\\zeta"},{"type":"folder","id":"16","title":"basic functions of complex numbers","collapsed":true},{"type":"text","id":"17","folderId":"16","text":"multiply"},{"type":"expression","id":"18","folderId":"16","color":"#c74440","latex":"M_{c}\\left(w,z\\right)=\\left(w.x\\cdot z.x-w.y\\cdot z.y,w.x\\cdot z.y+w.y\\cdot z.x\\right)"},{"type":"text","id":"19","folderId":"16","text":"reciprocate"},{"type":"expression","id":"20","folderId":"16","color":"#388c46","latex":"R_{c}\\left(z\\right)=\\frac{\\left(z.x,-z.y\\right)}{z.x^{2}+z.y^{2}}"},{"type":"text","id":"21","folderId":"16","text":"square"},{"type":"expression","id":"22","folderId":"16","color":"#2d70b3","latex":"Q_{c}\\left(z\\right)=M_{c}\\left(z,z\\right)"},{"type":"text","id":"23","folderId":"16","text":"cube"},{"type":"expression","id":"24","folderId":"16","color":"#000000","latex":"C_{c}\\left(z\\right)=M_{c}\\left(Q_{c}\\left(z\\right),z\\right)"},{"type":"text","id":"25","folderId":"16","text":"naturally exponentiate"},{"type":"expression","id":"26","folderId":"16","color":"#c74440","latex":"E_{c}\\left(z\\right)=\\left(1,0\\right)+z+\\frac{Q_{c}\\left(z\\right)}{2}+\\frac{C_{c}\\left(z\\right)}{6}+\\frac{Q_{c}\\left(Q_{c}\\left(z\\right)\\right)}{24}+\\frac{M_{c}\\left(Q_{c}\\left(z\\right),C_{c}\\left(z\\right)\\right)}{120}+\\frac{Q_{c}\\left(C_{c}\\left(z\\right)\\right)}{720}+\\frac{M_{c}\\left(Q_{c}\\left(Q_{c}\\left(z\\right)\\right),C_{c}\\left(z\\right)\\right)}{5040}+\\frac{Q_{c}\\left(Q_{c}\\left(Q_{c}\\left(z\\right)\\right)\\right)}{40320}"},{"type":"expression","id":"12","color":"#c74440","latex":"B=\\left[1,\\frac{1}{2},\\frac{1}{6},0,-\\frac{1}{30},0,\\frac{1}{42},0,-\\frac{1}{30},0,\\frac{5}{66},0,-\\frac{691}{2730}\\right]"},{"type":"expression","id":"5","color":"#6042a6","latex":"\\zeta\\left(s\\right)=M_{c}\\left(R_{c}\\left(s-\\left(1,0\\right)\\right),T\\left(\\left[\\frac{1}{n+1}\\cdot T\\left(\\left[\\left(-1\\right)^{k}\\cdot\\frac{n!}{k!\\left(n-k\\right)!}\\cdot R_{c}\\left(E_{c}\\left(\\left(s-\\left(1,0\\right)\\right)\\cdot\\ln\\left(k+1\\right)\\right)\\right)\\operatorname{for}k=\\left[0...n\\right]\\right]\\right)\\operatorname{for}n=\\left[0...N\\right]\\right]\\right)\\right)"},{"type":"expression","id":"31","color":"#000000","latex":"T\\left(v\\right)=\\left(\\sum_{k=1}^{\\operatorname{length}\\left(v\\right)}v\\left[k\\right].x,\\sum_{k=1}^{\\operatorname{length}\\left(v\\right)}v\\left[k\\right].y\\right)"},{"type":"expression","id":"33","color":"#c74440","latex":"\\zeta\\left(\\left(\\frac{1}{2},x\\right)\\right).x"},{"type":"expression","id":"32","color":"#c74440","latex":"N=10","slider":{"hardMin":true,"hardMax":true,"min":"0","max":"100","step":"1"}}]}}