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We consider the polygonal lines in the complex plane C whose N- th vertex is dened by SN = PN 0 n=0 exp(i!n 2 )( with ! 2 R), where the prime means that the rst and last terms in the sum are halved. By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small !, to Cornu spirals (C-spirals), we prove the precise renormalization formula N X k=0 0 exp(i!k2) exp(sgn(!)i=4) p j!j n X k=0

The approximate functional formula for the theta function and Diophantine Gauss sums