An interface for evaluation of Euler and related sums.
Implementation details:

EZ-Face is based on our fast method for computing Euler sums by turning them into convolutions of geometrically converging nested sums. This method is described in our preprint ``Special values of multidimensional polylogarithms,'' see the References page. The method was coded in the C language using the GNU MP multiprecision library. 

EZ-Face has been limited to the precision of 100 digits. A typical computation will complete within few seconds. 


Standard syntax for computer representation of mathematical expressions (as used by Maple) is assumed. 

The value $\zeta(s_1,\ldots,s_k)$ can be computed by typing in z( s1 , ... , sk )
The value $\zeta_{a + bi}(s_1,\ldots,s_k)$can be computed by typing in zp( a , b , s1 , ... , sk )
The value $\mu(s_1,\ldots,s_k)$can be computed by typing in m( s1 , ... , sk )
The value $\operatorname{mgl}(s_1,\ldots,s_k)$can be computed by typing in mgl( s1 , ... , sk )
The value $\operatorname{mcl}(s_1,\ldots,s_k)$can be computed by typing in mcl( s1 , ... , sk )

These functions can be used within any Maple expression. Their values are calculated before being passed to Maple. So, the arguments of these functions are not evaluated and must be non-zero integers (syntactically!), the only exception being the values of a and b, i.e. the first two arguments of zp(...), which may be integers or floating-point numbers. 

The function lindep( [ x1 , ... , xn ] ) can be used to discover a vanishing linear combination (with integer coefficients) of the values x1 , ... , xn.  These lindep expressions can either be entered in manually by hand or built by toggling the interface into lindep entry mode.  To enter a lindep expression using the lindep builder, first toggle the interface into the appropriate mode by selecting the option labeled Add to Lindep.  When the interface is in this mode, each expression entered into the text box labeled Expression will be added to the expressions collected in the box labeled Lindep.  Clicking on the button labeled Evaluate Lindep will evaluate the linear combination of these expressions.  In order to edit an expression already collected in the lindep builder, simply select that expression and click Delete Expression then re-add the expression into the lindep builder.  Lindep doesn't work on complex numbers, so to discover a relationship between complex numbers the real and imaginary parts should be done seperately.  Please note that all lindep calculations are performed with a significance of 100 digits.


Input Pi^6 / z(6) evaluates to 945.00000...
... Instance of Euler's formula for $\zeta(2n)$

Input z(3,1,3,1,3,1) - 2 * Pi^12 / 14! evaluates to 0
... Instance of the now proven Zagier conjecture.

Input mgl(1,1,2,1) - (Pi/3)^5/5!/2 evaluates to 0
... Instance of the proven formula for $\operatorname{mgl}(\{ 1 \}_b, 2, \{ 1 \}_a)$

Input lindep( [ z(-1,2) , log(2)*Pi^2 , z(3) ] ) evaluates to 12. , -1. , 3.
... Discovery of the relation $12\zeta(-1,2)-\log(2)\pi^2+3\zeta(3)=0$

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