stream This sounds very complicated – but let’s look at a couple of examples: This is a function which is defined for all values except for z = -1. I’m going to try and talk through another proof of this result. We use cookies to help provide and enhance our service and tailor content and ads. For the purposes of this post we will only look at real values of z (real numbers are still a subset of complex numbers). �Xxóy�K�O���8�5�M_(־� � �;@B�˚��f!�m0X�"����ѐ��%���C�(�/ߖ��a�r���w.Fw�0��)-'h$N�|/I�B����/��Z��x�-ŀuh�. Change ), You are commenting using your Twitter account. ( Log Out /  We also prove that the multiple Lucas zeta values at negative integer arguments are rational. x��\I��u�y��f��'w[�b�T�I�"lmT���>� �$ .�(���{YU�/�����3h��fw���-���˃�A�?����o�����͗72�r���r�p���8L��� �������l�b�Gw:k���~ 300 IB Maths Exploration ideas, video tutorials and Exploration Guides, November 9, 2014 in puzzles | Tags: analytic continuation, Riemann Zeta, Analytic Continuation and the Riemann Zeta Function. = -1/12 ? Many thanks! Advice on using Geogebra, Desmos and Tracker. There is also a fully typed up mark scheme. :�(��Z�^m ���-�ik���i����Q� Fourier Transforms – the most important tool in mathematics? ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Analytic continuation of the multiple Lucas zeta functions. my question is functional equation, how people developed functional equation ? is -1/12. Change ), You are commenting using your Google account. Blog at WordPress.com.Ben Eastaugh and Chris Sternal-Johnson. 6 0 obj <> We define the Hurwitz zeta function for with and with by see [3, 7–10, 12]. On analytic continuation of various multiple zeta-functions Kohji Matsumoto Abstract In this article we describe the development of the problem of analytic continuation of multiple zeta-functions. Proof of the R.H. © 2018 Elsevier Inc. All rights reserved. Through analytic continuation (where we extend the domain from z > 1 to all complex numbers apart from -1) we can rewrite the zeta function as: and substituting z = 2 into this formula, so that we end up with zeta(-1) we get: We have proved that 1 + 2 + 3 … = -1/12 ! For example, by using analytic continuation we can prove that the sum of the natural numbers (1 + 2 + 3 + ….) Analytic Continuation is a very important mathematical technique which allows us to extend the domain of functions. Unbelievable: 1+2+3+4…. (Don’t worry about how this is calculated – though it is related to the domain of convergence). However this formula for the zeta function is only valid for the domain z > 1, so we first need to extend the function through analytic continuation. A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Useful websites for use in the exploration, A selection of detailed exploration ideas. endobj However we can write a new function: Now, g(z) = f(z) for all z when z is not -1, but g(z) also exists when z = -1. But we can notice that the sum of a geometric sequence formula allows us to calculate f(z) in a different way: Here we have used the formula for summing a geometric, with the first term 1 and common ratio z. f(z) = g(z) when -1 < z< 1 , but g(z) is complex differentiable for all values except for z = 1 (when the denominator is 0). Copyright © 2020 Elsevier B.V. or its licensors or contributors. There are a few different methods to show this – some discussed previously here. Includes: Fourteen  full investigation questions – each one designed to last around 1 hour, and totaling around 35 pages and 500 marks worth of content. In this article, we consider more general situation. By continuing you agree to the use of cookies. This extends by analytic continuation the complex plane into the complex plane plus infinity. In particular, we prove the meromorphic continuation of the multiple Lucas zeta functions of depth d:∑00 for 1≤j≤d. Change ). Euler polynomials are related to the Hurwitz zeta function as We now consider the function as the analytic continuation of Euler numbers. �H{�4���Gw�^�����f���g� �p�+q_�]���}���:N�Gj��#g��Kr�_�����3J+p�I�՘}��@)4W���I��Q�n~�f�U��E�ix3� 絏=z����ы��G/e� ����DF��ΪT%�ֹu*�I������]��+���w�y��;�"3-��f��H��H����wy��f�B��wWMHa�sÃ�!�G��։2Y���o��q��$��&I�r��ӁkRoY��[Dԁ��6q�QG�IS?�G�&]��*����^��(�')s��s�����`r��+��j�M�m�s��K���[�=�F��h>�).z��f�š���)�f|ZǘR%���������iz�gz9cm�y��j�g��b�?a2endstream The proof revolves around the Riemann Zeta function, (Riemann is pictured above). We denote it by ζ (s). stream �{v�8��(�|�$B�8���������5E��W��"��H�S0Q^Ȧ������i{"������O�Qn�%������ A�-8V���|�ӯ��$�r�*��YQsR^=FL���E����M�v:16�uR~Q�٢�cY����K�n��X��CIc\�R�m��L�j���k�X�|�kpta��%Uech���8v�4E��9&i�5��g\�\&�[�@f�Ʊ �z�$�X����ذՃ"����@�?�R�2�Ī����5l,t�=��˼!�F�vZ��$��R���A6,�ȵ!螧k~���(Ğ��8*�Gq�b�E��I-&�$�y��@������BLC0�'}� a�H:�D8�v �`?+ ��[̌JS(&̋�;�N��1���20�d��Y�GP�Ȫ������)�Z6�$b�/ʏ�����d��j��s9R���>��.ǽ�Oi�Ϧ��D�H�b�(۸ȓ�H;]�v���IT��N�L[s��eJ�ֲ���� �x�5���1�g_�����yJ���@��?����@�)q��$;(��A�Oc��O;����/��oE`��׭���|tq^yD�����E2�MM���u� ���)�K\�q:A.&ϓ\��j���ĵ��迉�����Pn�ߴ�މ�(�gIPry9&�*'��i����)x�0 �n�־e����ډ���䃔��6_l^#�B���wY���!��u�:$��ps����7�=����������|�O����v�6v_L�+��|�} ��Ȋ9ݧ��|2��r�c�T*L �8��i�8�W��fB,��m6�⛋O5���hD2r0��z��z;�0�jɾ"����:5"RR�Fv'��������R>�>�?d���MS�%�P����� �_E$�$5}� o��3K99'�-\��@'S����v$�E;D�Vx_b�4�p�[�@.�$4����Ĺ"t�B�о^d ����yd0e��O�J�f\��/~_"�E��q3U����g���d�$El���b�7����'�U:��Z2ͽ3&����a���Y�tXc� Z�E��b��eN6;(ˉv��;�*{��N�;i���#�~1+TБW�����H�G( Therefore we can regard g(z) as the analytic continuation of f(z), and we have extended the domain of f(z) from all values except -1, to all values of z. On the other hand, the zeta function has an Euler product whose local factor at p … �o��Ў�x���HsI��qn��/�cG�X�r��jʍDBM0�3�C�-������[�tu�~��e�=�J�����\���7��-v�����׺X�EK�Hw\s��_khWУ��AI���W��e�D�-d-�2� e"y�Q�3���c-�9��]��]�U��JSi5��yWt��-+�CL0$:ٯaV;R��)���t���##��[8�#�`n�O_��QM)՟Zd��7�Z�����q�����՗��/�2_3u�qj�O�y���eKZ�q�$:J��h��E_��Ҷ��'����Q$dw��4��5����`�=:�N���ȍ�ꕓ_�|./������K]R]M-����x�H��s4���"e7�\�S����}%�� This is a continuation of our previous paper [7], in which multiple Fibonacci zeta functions of depth 2 have been studied.

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