סדרת פרי

הצגה של סדרת פרי עד F9

במתמטיקה, סדרת פרי (Farey) מסדר n היא סדרה של שברים מצומצמים בין 0 ו-1, שהמכנה שלהם הוא קטן או שווה ל-n, כאשר בכל סדרה האיבר הראשון הוא 0, האחרון הוא 1, האיבר האמצעי הוא 2\1 (n > 1) והסדרה נבנית לפי סדר עולה. ניתן לראות כי כל סדרת פרי מסדר n מכילה את כל הסדרות פרי מסדר הקטן מ-n. הסדרה מסדר n מסומנת על ידי Fn (ללא קשר לסדרת פיבונאצ'י)

סדרת פרי עד סדר 7 
 F1 = {0/1,                                                                                                          1/1}
 F2 = {0/1,                                                   1/2,                                                   1/1}
 F3 = {0/1,                               1/3,                1/2,                2/3,                               1/1}
 F4 = {0/1,                     1/4,      1/3,                1/2,                2/3,      3/4,                     1/1}
 F5 = {0/1,                1/5, 1/4,      1/3,      2/5,      1/2,      3/5,      2/3,      3/4, 4/5,                1/1}
 F6 = {0/1,           1/6, 1/5, 1/4,      1/3,      2/5,      1/2,      3/5,      2/3,      3/4, 4/5, 5/6,           1/1}
 F7 = {0/1,      1/7, 1/6, 1/5, 1/4, 2/7, 1/3,      2/5, 3/7, 1/2, 4/7, 3/5,      2/3, 5/7, 3/4, 4/5, 5/6, 6/7,      1/1}

מאפיינים

ניתן לראות כי אפשר לקשר בין כמות האיברים בין סדרות עולות על ידי פונקציית אוילר בצורה רקורסיבית (כאשר F1| = 2|):

או ללא התייחסות לפונקציית אוילר:

ועל ידי שימוש בנוסחת ההיפוך של מביוס נקבל:

מבחינה אסימפטוטית, ניתן לראות כי:

קשר להשערת רימן

אם נסמן את איברי הסדרה כ-, ונגדיר . בשנת 1924 הוכיח ג'רומה פראנל (Jérôme Franel) כי הטענה הבאה:

שקולה להשערת רימן, ובאותה שנה הוכיח אדמונד לנדאו כי הטענה:

גם היא שקולה להשערת רימן

קישורים חיצוניים

ויקישיתוף מדיה וקבצים בנושא סדרת פרי בוויקישיתוף

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