The Riemann zeta function
The Riemann zeta function ζ(s) is a function of a complex variable s = a + bi

Gamma


Dirichlet Eta Function Euler Product Gamma Function Infinite Product Reflection Formula
ETA Function Euler Product Gamma Function Infinite Product Reflection Formula
ETA Function Euler Product Gamma Function Infinite Product Reflection Formula
Wolfram MathWorld Wolfram MathWorld Wolfram MathWorld Wolfram MathWorld Wolfram MathWorld


Zeta Function Table (Simple Mode)

ζ(-20) = 1+20 + 2+20 + 3+20 + ... + n+20 = (0) + (1/21)*n21+(1/2)*n20+(1/12)(20)*n19+(0)n18-(1/120)(1140)*n17+(0)n16+(1/252)(15504)*n15+(0)n14-(1/240)(77520)*n13+(0)n12+(1/132)(167960)*n11+(0)n10-(691/32760)(167960)*n9 +(0)n8 +(1/12)(77520)*n7 +(0)n6 -(3617/8160)(15504)*n5 +(0)n4 +(43867/14364)(1140)*n3 +(0)n2 -(174611/6600)(20)*n1 +(0)n0
ζ(-19) = 1+19 + 2+19 + 3+19 + ... + n+19 = (174611/6600) + (1/20)*n20+(1/2)*n19+(1/12)(19)*n18+(0)n17-(1/120)( 969)*n16+(0)n15+(1/252)(11628)*n14+(0)n13-(1/240)(50388)*n12+(0)n11+(1/132)( 92378)*n10+(0)n9 -(691/32760)( 75582)*n8 +(0)n7 +(1/12)(27132)*n6 +(0)n5 -(3617/8160)( 3876)*n4 +(0)n3 +(43867/14364)( 171)*n2 +(0)n1 -(174611/6600)( 1)*n0
ζ(-18) = 1+18 + 2+18 + 3+18 + ... + n+18 = (0) + (1/19)*n19+(1/2)*n18+(1/12)(18)*n17+(0)n16-(1/120)( 816)*n15+(0)n14+(1/252)( 8568)*n13+(0)n12-(1/240)(31824)*n11+(0)n10+(1/132)( 48620)*n9 +(0)n8 -(691/32760)( 31824)*n7 +(0)n6 +(1/12)( 8568)*n5 +(0)n4 -(3617/8160)( 816)*n3 +(0)n2 +(43867/14364)( 18)*n1 +(0)n0
ζ(-17) = 1+17 + 2+17 + 3+17 + ... + n+17 = - (43867/14364) + (1/18)*n18+(1/2)*n17+(1/12)(17)*n16+(0)n15-(1/120)( 680)*n14+(0)n13+(1/252)( 6188)*n12+(0)n11-(1/240)(19448)*n10+(0)n9 +(1/132)( 24310)*n8 +(0)n7 -(691/32760)( 12376)*n6 +(0)n5 +(1/12)( 2380)*n4 +(0)n3 -(3617/8160)( 136)*n2 +(0)n1 +(43867/14364)( 1)*n0
ζ(-16) = 1+16 + 2+16 + 3+16 + ... + n+16 = (0) + (1/17)*n17+(1/2)*n16+(1/12)(16)*n15+(0)n14-(1/120)( 560)*n13+(0)n12+(1/252)( 4368)*n11+(0)n10-(1/240)(11440)*n9 +(0)n8 +(1/132)( 11440)*n7 +(0)n6 -(691/32760)( 4368)*n5 +(0)n4 +(1/12)( 560)*n3 +(0)n2 -(3617/8160)( 16)*n1 +(0)n0
ζ(-15) = 1+15 + 2+15 + 3+15 + ... + n+15 = (3617/8160) + (1/16)*n16+(1/2)*n15+(1/12)(15)*n14+(0)n13-(1/120)( 455)*n12+(0)n11+(1/252)( 3003)*n10+(0)n9 -(1/240)( 6435)*n8 +(0)n7 +(1/132)( 5005)*n6 +(0)n5 -(691/32760)( 1365)*n4 +(0)n3 +(1/12)( 105)*n2 +(0)n1 -(3617/8160)( 1)*n0
ζ(-14) = 1+14 + 2+14 + 3+14 + ... + n+14 = (0) + (1/15)*n15+(1/2)*n14+(1/12)(14)*n13+(0)n12-(1/120)( 364)*n11+(0)n10+(1/252)( 2002)*n9 +(0)n8 -(1/240)( 3432)*n7 +(0)n6 +(1/132)( 2002)*n5 +(0)n4 -(691/32760)( 364)*n3 +(0)n2 +(1/12)( 14)*n1 +(0)n0
ζ(-13) = 1+13 + 2+13 + 3+13 + ... + n+13 = - (1/12) + (1/14)*n14+(1/2)*n13+(1/12)(13)*n12+(0)n11-(1/120)( 286)*n10+(0)n9 +(1/252)( 1287)*n8 +(0)n7 -(1/240)( 1716)*n6 +(0)n5 +(1/132)( 715)*n4 +(0)n3 -(691/32760)( 78)*n2 +(0)n1 +(1/12)( 1)*n0
ζ(-12) = 1+12 + 2+12 + 3+12 + ... + n+12 = (0) + (1/13)*n13+(1/2)*n12+(1/12)(12)*n11+(0)n10-(1/120)( 220)*n9 +(0)n8 +(1/252)( 792)*n7 +(0)n6 -(1/240)( 792)*n5 +(0)n4 +(1/132)( 220)*n3 +(0)n2 -(691/32760)( 12)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (105n10 +525n9 +525n8 -1050n7 -1190n6 +2310n5 +1420n4 -3285n3 -287n2 +2073n-691)/455
ζ(-11) = 1+11 + 2+11 + 3+11 + ... + n+11 = (691/32760) + (1/12)*n12+(1/2)*n11+(1/12)(11)*n10+(0)n9 -(1/120)( 165)*n8 +(0)n7 +(1/252)( 462)*n6 +(0)n5 -(1/240)( 330)*n4 +(0)n3 +(1/132)( 55)*n2 +(0)n1 -(691/32760)( 1)*n0 = n(n+1)/2 * n(n+1)/2 * (2n8 +8n7 +4n6 -16n5 -5n4 +26n3 -3n2 -20n+10)/6
ζ(-10) = 1+10 + 2+10 + 3+10 + ... + n+10 = (0) + (1/11)*n11+(1/2)*n10+(1/12)(10)*n9 +(0)n8 -(1/120)( 120)*n7 +(0)n6 +(1/252)( 252)*n5 +(0)n4 -(1/240)( 120)*n3 +(0)n2 +(1/132)( 10)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (n2 +n-1) * (3n6 +9n5 +2n4 -11n3 +3n2 +10n-5)/11
ζ(-9) = 1+9 + 2+9 + 3+9 + ... + n+9 = - (1/132) + (1/10)*n10+(1/2)*n9 +(1/12)( 9)*n8 +(0)n7 -(1/120)( 84)*n6 +(0)n5 +(1/252)( 126)*n4 +(0)n3 -(1/240)( 36)*n2 +(0)n1 +(1/132)( 1)*n0 = n(n+1)/2 * n(n+1)/2 * (n2 +n-1) * (2n4 +4n3 -n2 -3n+3)/5
ζ(-8) = 1+8 + 2+8 + 3+8 + ... + n+8 = (0) + (1/9 )*n9 +(1/2)*n8 +(1/12)( 8)*n7 +(0)n6 -(1/120)( 56)*n5 +(0)n4 +(1/252)( 56)*n3 +(0)n2 -(1/240)( 8)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (5n6 +15n5 +5n4 -15n3 -n2 +9n-3)/15
ζ(-7) = 1+7 + 2+7 + 3+7 + ... + n+7 = (1/240) + (1/8 )*n8 +(1/2)*n7 +(1/12)( 7)*n6 +(0)n5 -(1/120)( 35)*n4 +(0)n3 +(1/252)( 21)*n2 +(0)n1 -(1/240)( 1)*n0 = n(n+1)/2 * n(n+1)/2 * (3n4 + 6n3 -n2 -4n-2)/6
ζ(-6) = 1+6 + 2+6 + 3+6 + ... + n+6 = (0) + (1/7 )*n7 +(1/2)*n6 +(1/12)( 6)*n5 +(0)n4 -(1/120)( 20)*n3 +(0)n2 +(1/252)( 6)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (3n4 + 6n3 -3n+1)/7
ζ(-5) = 1+5 + 2+5 + 3+5 + ... + n+5 = - (1/252) + (1/6 )*n6 +(1/2)*n5 +(1/12)( 5)*n4 +(0)n3 -(1/120)( 10)*n2 +(0)n1 +(1/252)( 1)*n0 = n(n+1)/2 * n(n+1)/2 * (2n2 + 2n-1)/3
ζ(-4) = 1+4 + 2+4 + 3+4 + ... + n+4 = (0) + (1/5 )*n5 +(1/2)*n4 +(1/12)( 4)*n3 +(0)n2 -(1/120)( 4)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3 * (3n2 + 3n-1)/5
ζ(-3) = 1+3 + 2+3 + 3+3 + ... + n+3 = (1/120) + (1/4 )*n4 +(1/2)*n3 +(1/12)( 3)*n2 +(0)n1 -(1/120)( 1)*n0 = n(n+1)/2 * n(n+1)/2
ζ(-2) = 1+2 + 2+2 + 3+2 + ... + n+2 = (0) + (1/3 )*n3 +(1/2)*n2 +(1/12)( 2)*n1 +(0)n0 = n(n+1)/2 * (2n+1)/3
ζ(-1) = 1+1 + 2+1 + 3+1 + ... + n+1 = - (1/12) + (1/2 )*n2 +(1/2)*n1 +(1/12)( 1)*n0 = n(n+1)/2
ζ( 0) = 10 + 20 + 30 + ... + n0 = - (1/2) + (1/1 )*n1 +(1/2)*n0 = n


Faulhaber's Formula My Correction For Bernoulli Numbers
Faulhabers_Formula Bernoulli_Numbers
Faulhabers_Formula Bernoulli_Numbers
Wolfram MathWorld Wolfram MathWorld


Zeta Function Table (Advanced Mode)

ζ(-20) = 1+20 + 2+20 + 3+20 + ... + n+20 = - (0) + (-1)(-(-1)!)((20)!/(20-(-1))!/(-1)!)*n20-(-1)+(1/2)*n20-0+(1/12)(20!/(20-1)!/1!)*n20-1+(0)n20-2+(-1)(1/120)(20!/(20-3)!/3!)*n20-3+(0)n20-4+(1/252)(20!/(20-5)!/5!)*n20-5+(0)n20-6+(-1)(1/240)(20!/(20-7)!/7!)*n20-7+(0)n20-8+(1/132)(20!/(20-9)!/9!)*n20-9+(0)n20-10+(-1)(691/32760)(20!/(20-11)!/11!)*n20-11+(0)n20-12+(1/12)(20!/(20-13)!/13!)*n20-13+(0)n20-14+(-1)(3617/8160)(20!/(20-15)!/15!)*n20-15+(0)n20-16+(43867/14364)(20!/(20-17)!/17!)*n20-17+(0)n20-18+(-1)(174611/6600)(20!/(20-19)!/19!)*n20-19+(0)n20-20
ζ(-19) = 1+19 + 2+19 + 3+19 + ... + n+19 = - (-1)(174611/6600) + (-1)(-(-1)!)((19)!/(19-(-1))!/(-1)!)*n19-(-1)+(1/2)*n19-0+(1/12)(19!/(19-1)!/1!)*n19-1+(0)n19-2+(-1)(1/120)(19!/(19-3)!/3!)*n19-3+(0)n19-4+(1/252)(19!/(19-5)!/5!)*n19-5+(0)n19-6+(-1)(1/240)(19!/(19-7)!/7!)*n19-7+(0)n19-8+(1/132)(19!/(19-9)!/9!)*n19-9+(0)n19-10+(-1)(691/32760)(19!/(19-11)!/11!)*n19-11+(0)n19-12+(1/12)(19!/(19-13)!/13!)*n19-13+(0)n19-14+(-1)(3617/8160)(19!/(19-15)!/15!)*n19-15+(0)n19-16+(43867/14364)(19!/(19-17)!/17!)*n19-17+(0)n19-18+(-1)(174611/6600)(19!/(19-19)!/19!)*n19-19
ζ(-18) = 1+18 + 2+18 + 3+18 + ... + n+18 = - (0) + (-1)(-(-1)!)((18)!/(18-(-1))!/(-1)!)*n18-(-1)+(1/2)*n18-0+(1/12)(18!/(18-1)!/1!)*n18-1+(0)n18-2+(-1)(1/120)(18!/(18-3)!/3!)*n18-3+(0)n18-4+(1/252)(18!/(18-5)!/5!)*n18-5+(0)n18-6+(-1)(1/240)(18!/(18-7)!/7!)*n18-7+(0)n18-8+(1/132)(18!/(18-9)!/9!)*n18-9+(0)n18-10+(-1)(691/32760)(18!/(18-11)!/11!)*n18-11+(0)n18-12+(1/12)(18!/(18-13)!/13!)*n18-13+(0)n18-14+(-1)(3617/8160)(18!/(18-15)!/15!)*n18-15+(0)n18-16+(43867/14364)(18!/(18-17)!/17!)*n18-17+(0)n18-18
ζ(-17) = 1+17 + 2+17 + 3+17 + ... + n+17 = - (43867/14364) + (-1)(-(-1)!)((17)!/(17-(-1))!/(-1)!)*n17-(-1)+(1/2)*n17-0+(1/12)(17!/(17-1)!/1!)*n17-1+(0)n17-2+(-1)(1/120)(17!/(17-3)!/3!)*n17-3+(0)n17-4+(1/252)(17!/(17-5)!/5!)*n17-5+(0)n17-6+(-1)(1/240)(17!/(17-7)!/7!)*n17-7+(0)n17-8+(1/132)(17!/(17-9)!/9!)*n17-9+(0)n17-10+(-1)(691/32760)(17!/(17-11)!/11!)*n17-11+(0)n17-12+(1/12)(17!/(17-13)!/13!)*n17-13+(0)n17-14+(-1)(3617/8160)(17!/(17-15)!/15!)*n17-15+(0)n17-16+(43867/14364)(17!/(17-17)!/17!)*n17-17
ζ(-16) = 1+16 + 2+16 + 3+16 + ... + n+16 = - (0) + (-1)(-(-1)!)((16)!/(16-(-1))!/(-1)!)*n16-(-1)+(1/2)*n16-0+(1/12)(16!/(16-1)!/1!)*n16-1+(0)n16-2+(-1)(1/120)(16!/(16-3)!/3!)*n16-3+(0)n16-4+(1/252)(16!/(16-5)!/5!)*n16-5+(0)n16-6+(-1)(1/240)(16!/(16-7)!/7!)*n16-7+(0)n16-8+(1/132)(16!/(16-9)!/9!)*n16-9+(0)n16-10+(-1)(691/32760)(16!/(16-11)!/11!)*n16-11+(0)n16-12+(1/12)(16!/(16-13)!/13!)*n16-13+(0)n16-14+(-1)(3617/8160)(16!/(16-15)!/15!)*n16-15+(0)n16-16
ζ(-15) = 1+15 + 2+15 + 3+15 + ... + n+15 = - (-1)(3617/8160) + (-1)(-(-1)!)((15)!/(15-(-1))!/(-1)!)*n15-(-1)+(1/2)*n15-0+(1/12)(15!/(15-1)!/1!)*n15-1+(0)n15-2+(-1)(1/120)(15!/(15-3)!/3!)*n15-3+(0)n15-4+(1/252)(15!/(15-5)!/5!)*n15-5+(0)n15-6+(-1)(1/240)(15!/(15-7)!/7!)*n15-7+(0)n15-8+(1/132)(15!/(15-9)!/9!)*n15-9+(0)n15-10+(-1)(691/32760)(15!/(15-11)!/11!)*n15-11+(0)n15-12+(1/12)(15!/(15-13)!/13!)*n15-13+(0)n15-14+(-1)(3617/8160)(15!/(15-15)!/15!)*n15-15
ζ(-14) = 1+14 + 2+14 + 3+14 + ... + n+14 = - (0) + (-1)(-(-1)!)((14)!/(14-(-1))!/(-1)!)*n14-(-1)+(1/2)*n14-0+(1/12)(14!/(14-1)!/1!)*n14-1+(0)n14-2+(-1)(1/120)(14!/(14-3)!/3!)*n14-3+(0)n14-4+(1/252)(14!/(14-5)!/5!)*n14-5+(0)n14-6+(-1)(1/240)(14!/(14-7)!/7!)*n14-7+(0)n14-8+(1/132)(14!/(14-9)!/9!)*n14-9+(0)n14-10+(-1)(691/32760)(14!/(14-11)!/11!)*n14-11+(0)n14-12+(1/12)(14!/(14-13)!/13!)*n14-13+(0)n14-14
ζ(-13) = 1+13 + 2+13 + 3+13 + ... + n+13 = - (1/12) + (-1)(-(-1)!)((13)!/(13-(-1))!/(-1)!)*n13-(-1)+(1/2)*n13-0+(1/12)(13!/(13-1)!/1!)*n13-1+(0)n13-2+(-1)(1/120)(13!/(13-3)!/3!)*n13-3+(0)n13-4+(1/252)(13!/(13-5)!/5!)*n13-5+(0)n13-6+(-1)(1/240)(13!/(13-7)!/7!)*n13-7+(0)n13-8+(1/132)(13!/(13-9)!/9!)*n13-9+(0)n13-10+(-1)(691/32760)(13!/(13-11)!/11!)*n13-11+(0)n13-12+(1/12)(13!/(13-13)!/13!)*n13-13
ζ(-12) = 1+12 + 2+12 + 3+12 + ... + n+12 = - (0) + (-1)(-(-1)!)((12)!/(12-(-1))!/(-1)!)*n12-(-1)+(1/2)*n12-0+(1/12)(12!/(12-1)!/1!)*n12-1+(0)n12-2+(-1)(1/120)(12!/(12-3)!/3!)*n12-3+(0)n12-4+(1/252)(12!/(12-5)!/5!)*n12-5+(0)n12-6+(-1)(1/240)(12!/(12-7)!/7!)*n12-7+(0)n12-8+(1/132)(12!/(12-9)!/9!)*n12-9+(0)n12-10+(-1)(691/32760)(12!/(12-11)!/11!)*n12-11+(0)n12-12
ζ(-11) = 1+11 + 2+11 + 3+11 + ... + n+11 = - (-1)(691/32760) + (-1)(-(-1)!)((11)!/(11-(-1))!/(-1)!)*n11-(-1)+(1/2)*n11-0+(1/12)(11!/(11-1)!/1!)*n11-1+(0)n11-2+(-1)(1/120)(11!/(11-3)!/3!)*n11-3+(0)n11-4+(1/252)(11!/(11-5)!/5!)*n11-5+(0)n11-6+(-1)(1/240)(11!/(11-7)!/7!)*n11-7+(0)n11-8+(1/132)(11!/(11-9)!/9!)*n11-9+(0)n11-10+(-1)(691/32760)(11!/(11-11)!/11!)*n11-11
ζ(-10) = 1+10 + 2+10 + 3+10 + ... + n+10 = - (0) + (-1)(-(-1)!)((10)!/(10-(-1))!/(-1)!)*n10-(-1)+(1/2)*n10-0+(1/12)(10!/(10-1)!/1!)*n10-1+(0)n10-2+(-1)(1/120)(10!/(10-3)!/3!)*n10-3+(0)n10-4+(1/252)(10!/(10-5)!/5!)*n10-5+(0)n10-6+(-1)(1/240)(10!/(10-7)!/7!)*n10-7+(0)n10-8+(1/132)(10!/(10-9)!/9!)*n10-9+(0)n10-10
ζ(-9) = 1+9 + 2+9 + 3+9 + ... + n+9 = - (1/132) + (-1)(-(-1)!)(( 9)!/( 9-(-1))!/(-1)!)*n 9-(-1)+(1/2)*n 9-0+(1/12)( 9!/( 9-1)!/1!)*n 9-1+(0)n 9-2+(-1)(1/120)( 9!/( 9-3)!/3!)*n 9-3+(0)n 9-4+(1/252)( 9!/( 9-5)!/5!)*n 9-5+(0)n 9-6+(-1)(1/240)( 9!/( 9-7)!/7!)*n 9-7+(0)n 9-8+(1/132)( 9!/( 9-9)!/9!)*n 9-9
ζ(-8) = 1+8 + 2+8 + 3+8 + ... + n+8 = - (0) + (-1)(-(-1)!)(( 8)!/( 8-(-1))!/(-1)!)*n 8-(-1)+(1/2)*n 8-0+(1/12)( 8!/( 8-1)!/1!)*n 8-1+(0)n 8-2+(-1)(1/120)( 8!/( 8-3)!/3!)*n 8-3+(0)n 8-4+(1/252)( 8!/( 8-5)!/5!)*n 8-5+(0)n 8-6+(-1)(1/240)( 8!/( 8-7)!/7!)*n 8-7+(0)n 8-8
ζ(-7) = 1+7 + 2+7 + 3+7 + ... + n+7 = - (-1)(1/240) + (-1)(-(-1)!)(( 7)!/( 7-(-1))!/(-1)!)*n 7-(-1)+(1/2)*n 7-0+(1/12)( 7!/( 7-1)!/1!)*n 7-1+(0)n 7-2+(-1)(1/120)( 7!/( 7-3)!/3!)*n 7-3+(0)n 7-4+(1/252)( 7!/( 7-5)!/5!)*n 7-5+(0)n 7-6+(-1)(1/240)( 7!/( 7-7)!/7!)*n 7-7
ζ(-6) = 1+6 + 2+6 + 3+6 + ... + n+6 = - (0) + (-1)(-(-1)!)(( 6)!/( 6-(-1))!/(-1)!)*n 6-(-1)+(1/2)*n 6-0+(1/12)( 6!/( 6-1)!/1!)*n 6-1+(0)n 6-2+(-1)(1/120)( 6!/( 6-3)!/3!)*n 6-3+(0)n 6-4+(1/252)( 6!/( 6-5)!/5!)*n 6-5+(0)n 6-6
ζ(-5) = 1+5 + 2+5 + 3+5 + ... + n+5 = - (1/252) + (-1)(-(-1)!)(( 5)!/( 5-(-1))!/(-1)!)*n 5-(-1)+(1/2)*n 5-0+(1/12)( 5!/( 5-1)!/1!)*n 5-1+(0)n 5-2+(-1)(1/120)( 5!/( 5-3)!/3!)*n 5-3+(0)n 5-4+(1/252)( 5!/( 5-5)!/5!)*n 5-5
ζ(-4) = 1+4 + 2+4 + 3+4 + ... + n+4 = - (0) + (-1)(-(-1)!)(( 4)!/( 4-(-1))!/(-1)!)*n 4-(-1)+(1/2)*n 4-0+(1/12)( 4!/( 4-1)!/1!)*n 4-1+(0)n 4-2+(-1)(1/120)( 4!/( 4-3)!/3!)*n 4-3+(0)n 4-4
ζ(-3) = 1+3 + 2+3 + 3+3 + ... + n+3 = - (-1)(1/120) + (-1)(-(-1)!)(( 3)!/( 3-(-1))!/(-1)!)*n 3-(-1)+(1/2)*n 3-0+(1/12)( 3!/( 3-1)!/1!)*n 3-1+(0)n 3-2+(-1)(1/120)( 3!/( 3-3)!/3!)*n 3-3
ζ(-2) = 1+2 + 2+2 + 3+2 + ... + n+2 = - (0) + (-1)(-(-1)!)(( 2)!/( 2-(-1))!/(-1)!)*n 2-(-1)+(1/2)*n 2-0+(1/12)( 2!/( 2-1)!/1!)*n 2-1+(0)n 2-2
ζ(-1) = 1+1 + 2+1 + 3+1 + ... + n+1 = - (1/12) + (-1)(-(-1)!)(( 1)!/( 1-(-1))!/(-1)!)*n 1-(-1)+(1/2)*n 1-0+(1/12)( 1!/( 1-1)!/1!)*n 1-1
ζ( 0) = 1 0 + 2 0 + 3 0 + ... + n 0 = - (1/2) + (-1)(-(-1)!)(( 0)!/( 0-(-1))!/(-1)!)*n 0-(-1)+(1/2)*n 0-0
ζ(+1) = 1-1 + 2-1 + 3-1 + ... + n-1 = - (-1)(-(-1 )!) + (-1)(-(-1)!)((-1)!/(-1-(-1))!/(-1)!)*n-1-(-1) + ln(n+1) - 1/(2n+2) + 0.57721566490153286060651209008240243104215933593992359880576723488486772677766467093694
ζ(+2) = 1-2 + 2-2 + 3-2 + ... = π+2 * 2+1 /( 1 )! * (1/12) = 1.64493406684822643647241516664602518921894990120679843773555822937000747040320087383362890061975870 ... A013661
ζ(+3) = 1-3 + 2-3 + 3-3 + ... = π+3 * 2+2 /( 2 )! * (0) + 1.20205690315959428539973816151144999076498629234049888179227155534183820578631309018645587360933525 = 1.20205690315959428539973816151144999076498629234049888179227155534183820578631309018645587360933525 ... A002117
ζ(+4) = 1-4 + 2-4 + 3-4 + ... = π+4 * 2+3 /( 3 )! * (1/120) = 1.08232323371113819151600369654116790277475095191872690768297621544412061618696884655690963594169991 ... A013662
ζ(+5) = 1-5 + 2-5 + 3-5 + ... = π+5 * 2+4 /( 4 )! * (0) + 1.03692775514336992633136548645703416805708091950191281197419267790380358978628148456004310655713333 = 1.03692775514336992633136548645703416805708091950191281197419267790380358978628148456004310655713333 ... A013663
ζ(+6) = 1-6 + 2-6 + 3-6 + ... = π+6 * 2+5 /( 5 )! * (1/252) = 1.01734306198444913971451792979092052790181749003285356184240866400433218290195789788277397793853517 ... A013664
ζ(+7) = 1-7 + 2-7 + 3-7 + ... = π+7 * 2+6 /( 6 )! * (0) + 1.00834927738192282683979754984979675959986356056523870641728313657160147831735573534609696891385132 = 1.00834927738192282683979754984979675959986356056523870641728313657160147831735573534609696891385132 ... A013665
ζ(+8) = 1-8 + 2-8 + 3-8 + ... = π+8 * 2+7 /( 7 )! * (1/240) = 1.00407735619794433937868523850865246525896079064985002032911020265258295257474881439528723037237197 ... A013666
ζ(+9) = 1-9 + 2-9 + 3-9 + ... = π+9 * 2+8 /( 8 )! * (0) + 1.00200839282608221441785276923241206048560585139488875654859661590978505339025839895039306912716958 = 1.00200839282608221441785276923241206048560585139488875654859661590978505339025839895039306912716958 ... A013667
ζ(+10) = 1-10 + 2-10 + 3-10 + ... = π+10 * 2+9 /( 9 )! * (1/132) = 1.00099457512781808533714595890031901700601953156447751725778899463629146515191295439704196861038565 ... A013668
ζ(+11) = 1-11 + 2-11 + 3-11 + ... = π+11 * 2+10 /( 10)! * (0) + 1.00049418860411946455870228252646993646860643575820861711914143610005405979821981470259184302356062 = 1.00049418860411946455870228252646993646860643575820861711914143610005405979821981470259184302356062 ... A013669
ζ(+12) = 1-12 + 2-12 + 3-12 + ... = π+12 * 2+11 /( 11)! * (691/32760) = 1.00024608655330804829863799804773967096041608845800340453304095213325201968194091304904280855190069 ... A013670
ζ(+13) = 1-13 + 2-13 + 3-13 + ... = π+13 * 2+12 /( 12)! * (0) + 1.00012271334757848914675183652635739571427510589550984513670267162089672682984420981289271395326813 = 1.00012271334757848914675183652635739571427510589550984513670267162089672682984420981289271395326813 ... A013671
ζ(+14) = 1-14 + 2-14 + 3-14 + ... = π+14 * 2+13 /( 13)! * (1/12) = 1.00006124813505870482925854510513533374748169616915454948275520225286294102317742087665978297199846 ... A013672
ζ(+15) = 1-15 + 2-15 + 3-15 + ... = π+15 * 2+14 /( 14)! * (0) + 1.00003058823630702049355172851064506258762794870685817750656993289333226715634227957307233434701754 = 1.00003058823630702049355172851064506258762794870685817750656993289333226715634227957307233434701754 ... A013673
ζ(+16) = 1-16 + 2-16 + 3-16 + ... = π+16 * 2+15 /( 15)! * (3617/8160) = 1.00001528225940865187173257148763672202323738899047153115310520358878708702795315178628560484632246 ... A013674
ζ(+17) = 1-17 + 2-17 + 3-17 + ... = π+17 * 2+16 /( 16)! * (0) + 1.00000763719763789976227360029356302921308824909026267909537984397293564329028245934208173863691667 = 1.00000763719763789976227360029356302921308824909026267909537984397293564329028245934208173863691667 ... A013675
ζ(+18) = 1-18 + 2-18 + 3-18 + ... = π+18 * 2+17 /( 17)! * (43867/14364) = 1.00000381729326499983985646164462193973045469721895333114317442998763003954265004563800196866898964 ... A013676
ζ(+19) = 1-19 + 2-19 + 3-19 + ... = π+19 * 2+18 /( 18)! * (0) + 1.00000190821271655393892565695779510135325857114483863023593304676182394970534130931266422711807630 = 1.00000190821271655393892565695779510135325857114483863023593304676182394970534130931266422711807630 ... A013677
ζ(+20) = 1-20 + 2-20 + 3-20 + ... = π+20 * 2+19 /( 19)! * (174611/6600) = 1.00000095396203387279611315203868344934594379418741059575005648985113751373114390025783609797638747 ... A013678

i did a little correction for Bernoulli Numbers emphasis on the first starting -1 index which is -(-1)!
that result i got by expanding the coefficient 1/(m+1) to the right form (-1)(B-1)(m!/(m-(-1))!/(-1)!)
i am also geting the right result of -1/2 by using the expanding result -(-1)! in the functional equation

i agree that the next step could be a controversial step but it seems to work ... that why i put it here
this is not a proof but only a way to show you that it seem to work as well. i know (-1)! is undefined!
but also Imaginary number were undefined at some point and now you are dividing imaginary numbers as well

Important Note: -(-1)! is not the sum of the harmonic series its the analytic continuation value of ζ(1)
(which is a nice way to show you a pole point)

My_Functional_Equation


Eta Function Table (Advanced Mode)

[Vixra] [PDF]

Here is my Advanced Mode Table:

η(-20) = 1+20 - 2+20 + 3+20 - ... ± n+20 = (0) + (-1)(n-1) * [ (1/2)*n20-0+(1/4)(20!/(20-1)!/1!)*n20-1+(0)n20-2+(-1)(1/8)(20!/(20-3)!/3!)*n20-3+(0)n20-4+(1/4)(20!/(20-5)!/5!)*n20-5+(0)n20-6+(-1)(17/16)(20!/(20-7)!/7!)*n20-7+(0)n20-8+(31/4)(20!/(20-9)!/9!)*n20-9+(0)n20-10+(-1)(691/8)(20!/(20-11)!/11!)*n20-11+(0)n20-12+(5461/4)(20!/(20-13)!/13!)*n20-13+(0)n20-14+(-1)(929569/32)(20!/(20-15)!/15!)*n20-15+(0)n20-16+(3202291/4)(20!/(20-17)!/17!)*n20-17+(0)n20-18+(-1)(221930581/8)(20!/(20-19)!/19!)*n20-19+(0)n20-20 ]
η(-19) = 1+19 - 2+19 + 3+19 - ... ± n+19 = (-1)(221930581/8) + (-1)(n-1) * [ (1/2)*n19-0+(1/4)(19!/(19-1)!/1!)*n19-1+(0)n19-2+(-1)(1/8)(19!/(19-3)!/3!)*n19-3+(0)n19-4+(1/4)(19!/(19-5)!/5!)*n19-5+(0)n19-6+(-1)(17/16)(19!/(19-7)!/7!)*n19-7+(0)n19-8+(31/4)(19!/(19-9)!/9!)*n19-9+(0)n19-10+(-1)(691/8)(19!/(19-11)!/11!)*n19-11+(0)n19-12+(5461/4)(19!/(19-13)!/13!)*n19-13+(0)n19-14+(-1)(929569/32)(19!/(19-15)!/15!)*n19-15+(0)n19-16+(3202291/4)(19!/(19-17)!/17!)*n19-17+(0)n19-18+(-1)(221930581/8)(19!/(19-19)!/19!)*n19-19 ]
η(-18) = 1+18 - 2+18 + 3+18 - ... ± n+18 = (0) + (-1)(n-1) * [ (1/2)*n18-0+(1/4)(18!/(18-1)!/1!)*n18-1+(0)n18-2+(-1)(1/8)(18!/(18-3)!/3!)*n18-3+(0)n18-4+(1/4)(18!/(18-5)!/5!)*n18-5+(0)n18-6+(-1)(17/16)(18!/(18-7)!/7!)*n18-7+(0)n18-8+(31/4)(18!/(18-9)!/9!)*n18-9+(0)n18-10+(-1)(691/8)(18!/(18-11)!/11!)*n18-11+(0)n18-12+(5461/4)(18!/(18-13)!/13!)*n18-13+(0)n18-14+(-1)(929569/32)(18!/(18-15)!/15!)*n18-15+(0)n18-16+(3202291/4)(18!/(18-17)!/17!)*n18-17+(0)n18-18 ]
η(-17) = 1+17 - 2+17 + 3+17 - ... ± n+17 = (3202291/4) + (-1)(n-1) * [ (1/2)*n17-0+(1/4)(17!/(17-1)!/1!)*n17-1+(0)n17-2+(-1)(1/8)(17!/(17-3)!/3!)*n17-3+(0)n17-4+(1/4)(17!/(17-5)!/5!)*n17-5+(0)n17-6+(-1)(17/16)(17!/(17-7)!/7!)*n17-7+(0)n17-8+(31/4)(17!/(17-9)!/9!)*n17-9+(0)n17-10+(-1)(691/8)(17!/(17-11)!/11!)*n17-11+(0)n17-12+(5461/4)(17!/(17-13)!/13!)*n17-13+(0)n17-14+(-1)(929569/32)(17!/(17-15)!/15!)*n17-15+(0)n17-16+(3202291/4)(17!/(17-17)!/17!)*n17-17 ]
η(-16) = 1+16 - 2+16 + 3+16 - ... ± n+16 = (0) + (-1)(n-1) * [ (1/2)*n16-0+(1/4)(16!/(16-1)!/1!)*n16-1+(0)n16-2+(-1)(1/8)(16!/(16-3)!/3!)*n16-3+(0)n16-4+(1/4)(16!/(16-5)!/5!)*n16-5+(0)n16-6+(-1)(17/16)(16!/(16-7)!/7!)*n16-7+(0)n16-8+(31/4)(16!/(16-9)!/9!)*n16-9+(0)n16-10+(-1)(691/8)(16!/(16-11)!/11!)*n16-11+(0)n16-12+(5461/4)(16!/(16-13)!/13!)*n16-13+(0)n16-14+(-1)(929569/32)(16!/(16-15)!/15!)*n16-15+(0)n16-16 ]
η(-15) = 1+15 - 2+15 + 3+15 - ... ± n+15 = (-1)(929569/32) + (-1)(n-1) * [ (1/2)*n15-0+(1/4)(15!/(15-1)!/1!)*n15-1+(0)n15-2+(-1)(1/8)(15!/(15-3)!/3!)*n15-3+(0)n15-4+(1/4)(15!/(15-5)!/5!)*n15-5+(0)n15-6+(-1)(17/16)(15!/(15-7)!/7!)*n15-7+(0)n15-8+(31/4)(15!/(15-9)!/9!)*n15-9+(0)n15-10+(-1)(691/8)(15!/(15-11)!/11!)*n15-11+(0)n15-12+(5461/4)(15!/(15-13)!/13!)*n15-13+(0)n15-14+(-1)(929569/32)(15!/(15-15)!/15!)*n15-15 ]
η(-14) = 1+14 - 2+14 + 3+14 - ... ± n+14 = (0) + (-1)(n-1) * [ (1/2)*n14-0+(1/4)(14!/(14-1)!/1!)*n14-1+(0)n14-2+(-1)(1/8)(14!/(14-3)!/3!)*n14-3+(0)n14-4+(1/4)(14!/(14-5)!/5!)*n14-5+(0)n14-6+(-1)(17/16)(14!/(14-7)!/7!)*n14-7+(0)n14-8+(31/4)(14!/(14-9)!/9!)*n14-9+(0)n14-10+(-1)(691/8)(14!/(14-11)!/11!)*n14-11+(0)n14-12+(5461/4)(14!/(14-13)!/13!)*n14-13+(0)n14-14 ]
η(-13) = 1+13 - 2+13 + 3+13 - ... ± n+13 = (5461/4) + (-1)(n-1) * [ (1/2)*n13-0+(1/4)(13!/(13-1)!/1!)*n13-1+(0)n13-2+(-1)(1/8)(13!/(13-3)!/3!)*n13-3+(0)n13-4+(1/4)(13!/(13-5)!/5!)*n13-5+(0)n13-6+(-1)(17/16)(13!/(13-7)!/7!)*n13-7+(0)n13-8+(31/4)(13!/(13-9)!/9!)*n13-9+(0)n13-10+(-1)(691/8)(13!/(13-11)!/11!)*n13-11+(0)n13-12+(5461/4)(13!/(13-13)!/13!)*n13-13 ]
η(-12) = 1+12 - 2+12 + 3+12 - ... ± n+12 = (0) + (-1)(n-1) * [ (1/2)*n12-0+(1/4)(12!/(12-1)!/1!)*n12-1+(0)n12-2+(-1)(1/8)(12!/(12-3)!/3!)*n12-3+(0)n12-4+(1/4)(12!/(12-5)!/5!)*n12-5+(0)n12-6+(-1)(17/16)(12!/(12-7)!/7!)*n12-7+(0)n12-8+(31/4)(12!/(12-9)!/9!)*n12-9+(0)n12-10+(-1)(691/8)(12!/(12-11)!/11!)*n12-11+(0)n12-12 ]
η(-11) = 1+11 - 2+11 + 3+11 - ... ± n+11 = (-1)(691/8) + (-1)(n-1) * [ (1/2)*n11-0+(1/4)(11!/(11-1)!/1!)*n11-1+(0)n11-2+(-1)(1/8)(11!/(11-3)!/3!)*n11-3+(0)n11-4+(1/4)(11!/(11-5)!/5!)*n11-5+(0)n11-6+(-1)(17/16)(11!/(11-7)!/7!)*n11-7+(0)n11-8+(31/4)(11!/(11-9)!/9!)*n11-9+(0)n11-10+(-1)(691/8)(11!/(11-11)!/11!)*n11-11 ]
η(-10) = 1+10 - 2+10 + 3+10 - ... ± n+10 = (0) + (-1)(n-1) * [ (1/2)*n10-0+(1/4)(10!/(10-1)!/1!)*n10-1+(0)n10-2+(-1)(1/8)(10!/(10-3)!/3!)*n10-3+(0)n10-4+(1/4)(10!/(10-5)!/5!)*n10-5+(0)n10-6+(-1)(17/16)(10!/(10-7)!/7!)*n10-7+(0)n10-8+(31/4)(10!/(10-9)!/9!)*n10-9+(0)n10-10 ]
η(-9) = 1+9 - 2+9 + 3+9 - ... ± n+9 = (31/4) + (-1)(n-1) * [ (1/2)*n 9-0+(1/4)( 9!/( 9-1)!/1!)*n 9-1+(0)n 9-2+(-1)(1/8)( 9!/( 9-3)!/3!)*n 9-3+(0)n 9-4+(1/4)( 9!/( 9-5)!/5!)*n 9-5+(0)n 9-6+(-1)(17/16)( 9!/( 9-7)!/7!)*n 9-7+(0)n 9-8+(31/4)( 9!/(10-9)!/9!)*n 9-9 ]
η(-8) = 1+8 - 2+8 + 3+8 - ... ± n+8 = (0) + (-1)(n-1) * [ (1/2)*n 8-0+(1/4)( 8!/( 8-1)!/1!)*n 8-1+(0)n 8-2+(-1)(1/8)( 8!/( 8-3)!/3!)*n 8-3+(0)n 8-4+(1/4)( 8!/( 8-5)!/5!)*n 8-5+(0)n 8-6+(-1)(17/16)( 8!/( 8-7)!/7!)*n 8-7+(0)n 8-8 ]
η(-7) = 1+7 - 2+7 + 3+7 - ... ± n+7 = (-1)(17/16) + (-1)(n-1) * [ (1/2)*n 7-0+(1/4)( 7!/( 7-1)!/1!)*n 7-1+(0)n 7-2+(-1)(1/8)( 7!/( 7-3)!/3!)*n 7-3+(0)n 7-4+(1/4)( 7!/( 7-5)!/5!)*n 7-5+(0)n 7-6+(-1)(17/16)( 7!/( 7-7)!/7!)*n 7-7 ]
η(-6) = 1+6 - 2+6 + 3+6 - ... ± n+6 = (0) + (-1)(n-1) * [ (1/2)*n 6-0+(1/4)( 6!/( 6-1)!/1!)*n 6-1+(0)n 6-2+(-1)(1/8)( 6!/( 6-3)!/3!)*n 6-3+(0)n 6-4+(1/4)( 6!/( 6-5)!/5!)*n 6-5+(0)n 6-6 ]
η(-5) = 1+5 - 2+5 + 3+5 - ... ± n+5 = (1/4) + (-1)(n-1) * [ (1/2)*n 5-0+(1/4)( 5!/( 5-1)!/1!)*n 5-1+(0)n 5-2+(-1)(1/8)( 5!/( 5-3)!/3!)*n 5-3+(0)n 5-4+(1/4)( 5!/( 5-5)!/5!)*n 5-5 ]
η(-4) = 1+4 - 2+4 + 3+4 - ... ± n+4 = (0) + (-1)(n-1) * [ (1/2)*n 4-0+(1/4)( 4!/( 4-1)!/1!)*n 4-1+(0)n 4-2+(-1)(1/8)( 4!/( 4-3)!/3!)*n 4-3+(0)n 4-4 ]
η(-3) = 1+3 - 2+3 + 3+3 - ... ± n+3 = (-1)(1/8) + (-1)(n-1) * [ (1/2)*n 3-0+(1/4)( 3!/( 3-1)!/1!)*n 3-1+(0)n 3-2+(-1)(1/8)( 3!/( 3-3)!/3!)*n 3-3 ]
η(-2) = 1+2 - 2+2 + 3+2 - ... ± n+2 = (0) + (-1)(n-1) * [ (1/2)*n 2-0+(1/4)( 2!/( 2-1)!/1!)*n 2-1+(0)n 2-2 ]
η(-1) = 1+1 - 2+1 + 3+1 - ... ± n+1 = (1/4) + (-1)(n-1) * [ (1/2)*n 1-0+(1/4)( 1!/( 1-1)!/1!)*n 1-1 ]
η( 0) = 1 0 - 2 0 + 3 0 - ... ± n 0 = (1/2) + (-1)(n-1) * [ (1/2)*n 0-0 ]
η(+1) = 1-1 - 2-1 + 3-1 - ... = π+1 * (1-2 0 )/(1-2+1 ) /( 0 )! * (1/2) + ln(2)
η(+2) = 1-2 - 2-2 + 3-2 - ... = π+2 * (1-2+1 )/(1-2+2 ) /( 1 )! * (1/4)
η(+3) = 1-3 - 2-3 + 3-3 - ... = π+3 * (1-2+2 )/(1-2+3 ) /( 2 )! * (0) + 0.9015426773696957140498036211335874930737397192553741613442036665063786543397348176398419052070014436096493683464455395638689969 ... A197070
η(+4) = 1-4 - 2-4 + 3-4 - ... = π+4 * (1-2+3 )/(1-2+4 ) /( 3 )! * (1/8)
η(+5) = 1-5 - 2-5 + 3-5 - ... = π+5 * (1-2+4 )/(1-2+5 ) /( 4 )! * (0) + 0.9721197704469093059356551435534695325535133620330432612258056355348158654246388917750404123973125028558940701248968209776259016 ... A267316
η(+6) = 1-6 - 2-6 + 3-6 - ... = π+6 * (1-2+5 )/(1-2+6 ) /( 5 )! * (1/4)
η(+7) = 1-7 - 2-7 + 3-7 - ... = π+7 * (1-2+6 )/(1-2+7 ) /( 6 )! * (0) + 0.9925938199228302826704257131333936852311156924314068516295130875626702052186470519813142037745723970319464309381865268421729215 ... A275710
η(+8) = 1-8 - 2-8 + 3-8 - ... = π+8 * (1-2+7 )/(1-2+8 ) /( 7 )! * (17/16)
η(+9) = 1-9 - 2-9 + 3-9 - ... = π+9 * (1-2+8 )/(1-2+9 ) /( 8 )! * (0) + 0.9980942975416053307677830318525979508743339535378774723433286603788874555254527020794930962008915799614812274034804701822791811 ... A347059
η(+10) = 1-10 - 2-10 + 3-10 - ... = π+10 * (1-2+9 )/(1-2+10) /( 9 )! * (31/4)
η(+11) = 1-11 - 2-11 + 3-11 - ... = π+11 * (1-2+10)/(1-2+11) /( 10)! * (0) + 0.9995171434980607541440941748286901806712738122857884915164860245413626007554481156647963431768579339079323365487778793663825129 ... No OEIS
η(+12) = 1-12 - 2-12 + 3-12 - ... = π+12 * (1-2+11)/(1-2+12) /( 11)! * (691/8)
η(+13) = 1-13 - 2-13 + 3-13 - ... = π+13 * (1-2+12)/(1-2+13) /( 12)! * (0) + 0.9998785427632651154921749928162679529907120504497345741784173438202080313398955173788563631930256384356203665086284936791806474 ... No OEIS
η(+14) = 1-14 - 2-14 + 3-14 - ... = π+14 * (1-2+13)/(1-2+14) /( 13)! * (5461/4)
η(+15) = 1-15 - 2-15 + 3-15 - ... = π+15 * (1-2+14)/(1-2+15) /( 14)! * (0) + 0.9999695512130992380826329326287779577864446218056919874322592291620765706068332254788601106937981260365731228040854496786124765 ... No OEIS
η(+16) = 1-16 - 2-16 + 3-16 - ... = π+16 * (1-2+15)/(1-2+16) /( 15)! * (929569/32)
η(+17) = 1-17 - 2-17 + 3-17 - ... = π+17 * (1-2+16)/(1-2+17) /( 16)! * (0) + 0.9999923782920410119769378722418007372967489380513056133196368114435781461033426051785785940791371772418412846033633549909106784 ... No OEIS
η(+18) = 1-18 - 2-18 + 3-18 - ... = π+18 * (1-2+17)/(1-2+18) /( 17)! * (3202291/4)
η(+19) = 1-19 - 2-19 + 3-19 - ... = π+19 * (1-2+18)/(1-2+19) /( 18)! * (0) + 0.9999980935081716751068564929652682542955294098496301004254844538775818468002597307096471347405009316270705371385138420843806958 ... No OEIS
η(+20) = 1-20 - 2-20 + 3-20 - ... = π+20 * (1-2+19)/(1-2+20) /( 19)! * (221930581/8)


Picturing the Zeta Function on the Complex plane

1.1_i1.8

A Deeper Understanding : "origin points"

Many people are using the term "Assigned Value" or "Analytic Continuation" for divergent series
But this explanation is so lacking and can be replaced with a much easier and simpler term of explanation

For me (as I see it) when I am looking at the zeta function I dont see (or use) the term "Assigned Value" or "Analytic Continuation"
Instead I see "spirals" all around the grid!

The simplest way is to first look at the Complex plane ζ(s)=ζ(x+iy)=a+ib where Re(s)>1 and the behavior of convergent points (above spiral picture!)
The spiral swirls around inwards to an unique point which the series Converges - Same goes for the other way around!

When I look at the Complex plane ζ(s)=ζ(x+iy)=a+ib where Re(s)<1 and the behavior of divergent points
The spiral swirls around outwards but if you look closely you will notice that the spiral has a "center point" or an "origin"
and that "origin" is the "Assigned Value" everyone is talking about

when I first started to read about the zeta function I didn’t know what are those "Assigned Values" or "Analytic Continuation"
and how and why people are trying to give a value for divergent series And why that specific value and not something else?
I wanted an explanation other then "because the formula says so" and without going deeper into all the "Analytic Continuation stuff".

Those "origin points" did the trick!

the simplest origin point to understend is η(-1)=1-2+3-4+5-6+...
Origin_Point

the (Assigned) value 1/4 is not the summation of η(-1)
it's simply represents the intersection points of the two lines
or as i like to describe it as the origin point of the spiral on the complex plane

make sure to check my article that i submitted to Vixra >>> [PDF]

If you are assigning a value for a series that decreases to a specific value (case #1)
Then you can assigning a value for a series that increases from a specific value (case #2) <<< origin point!

Other then those two cases there is one more
This is when the spiral at some point start to spin around a specific value with a "fixed radius"
those cases appears at the zeta function ζ(s)=ζ(x+iy)=a+ib when x=1 and the radius will be 1/y
meaning that this is a divergent series with a "fixed radius"

Its true that the zeta function spirals have 3 cases but they are all spirals with one arm
Now at the eta function the spirals have two arms (that is because of the +/- swapping) with the same 3 cases

By the way the "fixed radius" appears at the eta function η(s)=η(x+iy)=a+ib when Re(s)=0


ζ(s)=ζ(1.1+1.8i) function - spiral with one arm - convergent case , Re(s)>1 ζ(s)=ζ(1+1.8i) function - spiral with one arm - fixed radius case , Re(s)=1 ζ(s)=ζ(0.6+1.8i) function - spiral with one arm - divergent case , Re(s)<1
η(s)=η(0.2+2i) function - spiral with two arms - convergent case , Re(s)>0 η(s)=η(0+2i) function - spiral with two arms - fixed radius case , Re(s)=0 η(s)=η(-0.4+2i) function - spiral with two arms - divergent case , Re(s)<0




Removing the Riemann Hypothesis from the Complex Plane

you dont need the complex plane or the use of analytic continuation in order to see or understand the riemann hypothesis! here is the eta function rotation on the complex plane.

as you can see when ever zeta is getting an "analytic continuation zero" then the eta tunction is also getting a zero but this is a "real zero" because eta is well defined for re(s)>0

same goes when we take eta function summation of the x and y Axis when they are both equal to zero. you dont need the complex plane anymore

Analytic Continuation Zero "Real Zero"
ζ(s)=ζ(0.5+14.1347251417i) function - spiral with one arm - divergent zero η(s)=η(0.5+14.1347251417i) function - spiral with two arms - convergent zero






The Riemann Hypothesis Proof

[Vixra] [PDF] My Email: isaac.mor@hotmail.com



Riemann zeta function formula

[Vixra] [PDF] My Email: isaac.mor@hotmail.com

Riemann_zeta_function_formula



Valid XHTML Valid CSS