By Scott Malone

BOSTON (Reuters) - U.S. prosecutors on Tuesday rejected a claim by the Boston Marathon bomber's lawyers that they had not proven that the death of the attack's youngest victim took a heavy enough toll on his family to influence a jury to sentence bomber Dzhokhar Tsarnaev to death.

The prosecution's statement, in a filing in U.S. District Court in Boston, comes a day before the two sides are to give their closing arguments. After those arguments, the same jury that found the 21-year-old ethnic Chechen guilty of killing three people in the April 15, 2013, attack begin deliberations on whether to sentence to Tsarnaev to death by lethal injection or to life in prison without possibility of release.

On Monday, defense attorneys had argued that prosecutors had not shown evidence to back up the claim that the death of 8-year-old Martin Richard was one of the "aggravating factors" that the jury could take into account in making its decision on Tsarnaev's fate.

Richard's father, William, during the first phase of the trial described being thrown through the air by the twin pressure cooker bombs, seeing his three children's injuries and making the gut-wrenching decision to leave badly wounded Martin at the scene with his wife, who was also injured, so that he could save the life of his youngest daughter, Jane.

None of the Martin family was called to testify in the trial's sentencing phase after they asked federal prosecutors in a statement published on the front page of the Boston Globe newspaper to drop their quest for the death penalty.

Instead they heard from a doctor who described the enormous injuries Martin Richard had sustained.

"The injuries to Martin covered his entire body and were described as extraordinarily painful," prosecutors wrote in a court filing on Tuesday. "This evidence comports with the type of "victim impact" evidence which is clearly admissible."

In making their sentencing decision, the jury is expected to weigh the "aggravating" factors of Tsarnaev's crime, including the young age of some of his victims and the large number of people injured, against "mitigating" factors put forth by the defense, which contends Tsarnaev played a secondary role in an attack planned and driven by his 26-year-old brother, Tamerlan.

Tamerlan died four days after the bombing following a gunfight with police that ended when Dzohkhar ran him over with a stolen car.

(Reporting by Scott Malone; editing by Andrew Hay)