This somewhat explains it: In[59]:= FullSimplify[FunctionExpand[LerchPhi[1, s, a]/Zeta[s, a]], s > 1 && a > 0] Out[59]= LerchPhi[1, s, a]/Zeta[s, a] In[60]:= Table[%, {s, 2, 3, 1/2}] Out[60]= {Zeta[2, a, IncludeSingularTerm -> False]/Zeta[2, a], Zeta[5/2, a, IncludeSingularTerm -> False]/Zeta[5/2, a], Zeta[3, a, IncludeSingularTerm -> False]/Zeta[3, a]} In[61]:= FullSimplify[FunctionExpand[%], a > 0] Out[61]= {1, 1, 1} but I don't see why I didn't get this for %59. --rwg Wouter> It is quite irresistible to play around with these formulae until one finds a simple expression that even Mma 9.0.0 can't simplify: try this: LerchPhi[1, 2*k, 1/2]/2^(2*k) /( -(-1)^k BernoulliB[2k] (2 Pi)^(2k)/(2k)!/2) == 1-1/2^(2k) or shorter: FullSimplify[LerchPhi[1, 2*k, 1/2]/2^(2*k)/Zeta[2 k]] just cute, no? Wouter -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Simon Plouffe Sent: woensdag 13 maart 2013 23:02 To: math-fun Subject: [math-fun] a certain formula of Euler, primes and Pi Hello, I have a friend in Moncton (NB, canada) a math professor, Paul Deguire, browsing around with the history of math and came accros this formula (well known ?) of pi : http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80 see the center of the page with prime numbers, also this one http://mathworld.wolfram.com/PiFormulas.html formulas 60 and 61. It deals with prime numbers , an infinite product and pi. Is there someone that knows a reference for this formula, when it was found by Euler ? Any information that would enlight this almost pi day, Best regards, Simon Plouffe