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Poussin proof

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proved that if a large number n is divided by all the primes up to n, then the average fraction by which the quotient falls short of the next whole number is Îł:
394: 49: 154: 40: 418: 387: 147: 380: 262: 413: 275: 239:{\displaystyle {\frac {\sum _{p\leq n}\left\{{\frac {n}{p}}\right\}}{\pi (n)}}\approx 1-\gamma ,} 43:
proved an approximate formula for the average number of divisors of all the numbers from 1 to n:
329: 326: 307: 143: 254: 29: 364: 407: 360: 310: 295:(1898), pp. 84–90. Cited in MathWorld article "Euler-Mascheroni Constant" below. 17: 352: 265:. For example, if we divide 29 by 2, we get 14.5, which falls short of 15 by 0.5. 21: 285:(1838), pp. 259–274. Cited in MathWorld article "Divisor Function" below. 334: 315: 128:{\displaystyle {\frac {\sum _{k=1}^{n}d(k)}{n}}\approx \ln n+2\gamma -1,} 33: 276:
Sur l'usage des séries infinies dans la théorie des nombres
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de la Vallée Poussin, C.-J. Untitled communication.
238: 127: 290:Annales de la Societe Scientifique de Bruxelles 280:Journal fĂĽr die reine und angewandte Mathematik 388: 8: 28:is the proof of an identity related to the 395: 381: 190: 174: 167: 165: 71: 60: 53: 51: 7: 349: 347: 14: 155:Charles Jean de la VallĂ©e-Poussin 351: 215: 209: 86: 80: 41:Peter Gustav Lejeune Dirichlet 1: 367:. You can help Knowledge by 330:"Euler-Mascheroni Constant" 435: 346: 148:Euler-Mascheroni constant 263:prime-counting function 261:, and Ď€ represents the 146:, and Îł represents the 363:-related article is a 240: 129: 76: 241: 130: 56: 164: 50: 419:Number theory stubs 327:Weisstein, Eric W. 311:"Divisor Function" 308:Weisstein, Eric W. 274:Dirichlet, G. L. " 236: 185: 125: 376: 375: 253:} represents the 219: 198: 170: 93: 426: 397: 390: 383: 355: 348: 340: 339: 321: 320: 245: 243: 242: 237: 220: 218: 204: 203: 199: 191: 184: 168: 144:divisor function 134: 132: 131: 126: 94: 89: 75: 70: 54: 434: 433: 429: 428: 427: 425: 424: 423: 404: 403: 402: 401: 344: 325: 324: 306: 305: 302: 271: 255:fractional part 205: 186: 169: 162: 161: 142:represents the 55: 48: 47: 30:fractional part 12: 11: 5: 432: 430: 422: 421: 416: 406: 405: 400: 399: 392: 385: 377: 374: 373: 356: 342: 341: 322: 301: 300:External links 298: 297: 296: 286: 270: 267: 247: 246: 235: 232: 229: 226: 223: 217: 214: 211: 208: 202: 197: 194: 189: 183: 180: 177: 173: 136: 135: 124: 121: 118: 115: 112: 109: 106: 103: 100: 97: 92: 88: 85: 82: 79: 74: 69: 66: 63: 59: 20:, a branch of 13: 10: 9: 6: 4: 3: 2: 431: 420: 417: 415: 414:Number theory 412: 411: 409: 398: 393: 391: 386: 384: 379: 378: 372: 370: 366: 362: 361:number theory 357: 354: 350: 345: 337: 336: 331: 328: 323: 318: 317: 312: 309: 304: 303: 299: 294: 291: 287: 284: 281: 277: 273: 272: 268: 266: 264: 260: 256: 252: 233: 230: 227: 224: 221: 212: 206: 200: 195: 192: 187: 181: 178: 175: 171: 160: 159: 158: 156: 151: 149: 145: 141: 122: 119: 116: 113: 110: 107: 104: 101: 98: 95: 90: 83: 77: 72: 67: 64: 61: 57: 46: 45: 44: 42: 37: 35: 31: 27: 26:Poussin proof 23: 19: 18:number theory 369:expanding it 358: 343: 333: 314: 292: 289: 282: 279: 258: 250: 248: 152: 139: 137: 38: 25: 15: 22:mathematics 408:Categories 269:References 335:MathWorld 316:MathWorld 231:γ 228:− 222:≈ 207:π 179:≤ 172:∑ 153:In 1898, 117:− 114:γ 102:⁡ 96:≈ 58:∑ 39:In 1838, 249:where { 138:where 24:, the 359:This 34:ratio 32:of a 365:stub 278:", 257:of 16:In 410:: 332:. 313:. 293:22 283:18 150:. 99:ln 36:. 396:e 389:t 382:v 371:. 338:. 319:. 259:x 251:x 234:, 225:1 216:) 213:n 210:( 201:} 196:p 193:n 188:{ 182:n 176:p 140:d 123:, 120:1 111:2 108:+ 105:n 91:n 87:) 84:k 81:( 78:d 73:n 68:1 65:= 62:k

Index

number theory
mathematics
fractional part
ratio
Peter Gustav Lejeune Dirichlet
divisor function
Euler-Mascheroni constant
Charles Jean de la Vallée-Poussin
fractional part
prime-counting function
Sur l'usage des séries infinies dans la théorie des nombres
Weisstein, Eric W.
"Divisor Function"
MathWorld
Weisstein, Eric W.
"Euler-Mascheroni Constant"
MathWorld
Stub icon
number theory
stub
expanding it
v
t
e
Categories
Number theory
Number theory stubs

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